DOT/FRA/ORD-23/21 Final Report | August 2023

Measurement and Characterization of Track Geometry 

Data: Literature Review and Recommendations for 

Processing FRA ATIP Program Data

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Measurement and Characterization of Track Geometry Data: Literature Review 

and Recommendations for Processing FRA ATIP Program Data

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Alexander Keylin: ORCID 0000-0003-2786-3992

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Transportation Technology Center, Inc. (TTCI)

55500 DOT Rd.

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U.S. Department of Transportation 

Federal Railroad Administration

Office of Railroad Policy and Development

Office of Research, Development, and Technology (RD&T)

Washington, DC 20590

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13. ABSTRACT (Maximum 200 words)

From October 2018 to March 2019, the Federal Railroad Administration sponsored Transportation Technology Center, Inc. to 

conduct a literature review on the methods of measurement and characterization of track geometry. The goal of the review was to 

summarize the current state of track geometry measurement and to provide recommendations on methods for processing and 

characterizing track geometry data collected under FRA’s Automated Track Inspection Program.

14. SUBJECT TERMS

Track geometry, track quality, track irregularities, space curve, Track Quality Index, TQI, Track 

Geometry Measurement System, TGMS, Automated Track Inspection Program, ATIP, Power 

Spectral Density, PSD.

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106 

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Standard Form 298 (Rev. 8/98)

Prescribed by ANSI Std. Z39.18
ii

METRIC/ENGLISH CONVERSION FACTORS

ENGLISH TO METRIC METRIC TO ENGLISH

LENGTH (APPROXIMATE) LENGTH (APPROXIMATE) 

1 inch (in) = 2.5 centimeters (cm) 1 millimeter (mm) = 0.04 inch (in)

1 foot (ft) = 30 centimeters (cm) 1 centimeter (cm) = 0.4 inch (in)

1 yard (yd) = 0.9 meter (m) 1 meter (m) = 3.3 feet (ft)

1 mile (mi) = 1.6 kilometers (km) 1 meter (m) = 1.1 yards (yd)

1 kilometer (km) = 0.6 mile (mi)

AREA (APPROXIMATE) AREA (APPROXIMATE) 

1 square inch (sq in, in2) = 6.5 square centimeters (cm2) 1 square centimeter (cm2) = 0.16 square inch (sq in, in2) 

1 square foot (sq ft, ft2) = 0.09 square meter (m2) 1 square meter (m2) = 1.2 square yards (sq yd, yd2)

1 square yard (sq yd, yd2) = 0.8 square meter (m2) 1 square kilometer (km2) = 0.4 square mile (sq mi, mi2)

1 square mile (sq mi, mi2) = 2.6 square kilometers (km2) 10,000 square meters (m2) = 1 hectare (ha) = 2.5 acres

1 acre = 0.4 hectare (he) = 4,000 square meters (m2) 

MASS - WEIGHT (APPROXIMATE) MASS - WEIGHT (APPROXIMATE) 

1 ounce (oz) = 28 grams (gm) 1 gram (gm) = 0.036 ounce (oz)

1 pound (lb) = 0.45 kilogram (kg) 1 kilogram (kg) = 2.2 pounds (lb)

1 short ton = 2,000 pounds (lb) = 0.9 tonne (t) 1 tonne (t) =

=

1,000 kilograms (kg)

1.1 short tons

VOLUME (APPROXIMATE) VOLUME (APPROXIMATE) 

1 teaspoon (tsp) = 5 milliliters (ml) 1 milliliter (ml) = 0.03 fluid ounce (fl oz)

1 tablespoon (tbsp) = 15 milliliters (ml) 1 liter (l) = 2.1 pints (pt)

1 fluid ounce (fl oz) = 30 milliliters (ml) 1 liter (l) = 1.06 quarts (qt)

1 cup (c) = 0.24 liter (l) 1 liter (l) = 0.26 gallon (gal)

1 pint (pt) = 0.47 liter (l)

1 quart (qt) = 0.96 liter (l)

1 gallon (gal) = 3.8 liters (l)

1 cubic foot (cu ft, ft3) = 0.03 cubic meter (m3) 1 cubic meter (m3) = 36 cubic feet (cu ft, ft3) 

1 cubic yard (cu yd, yd3) = 0.76 cubic meter (m3) 1 cubic meter (m3) = 1.3 cubic yards (cu yd, yd3)

TEMPERATURE (EXACT) TEMPERATURE (EXACT) 

[(x-32)(5/9)] °F = y °C [(9/5) y + 32] °C = x °F

QUICK INCH - CENTIMETER LENGTH CONVERSION

0 1 2 3 4 5

Inches

Centimeters 0 1 2 3 4 5 6 987 1110 1312

QUICK FAHRENHEIT - CELSIUS TEMPERATURE CONVERSIO

 -40° -22° -4° 14° 32° 50° 68° 86° 104° 122° 140° 158° 176° 194° 212°

 

°F

 °C -40° -30° -20° -10° 0° 10° 20° 30° 40° 50° 60° 70° 80° 90° 100°

For more exact and or other conversion factors, see NIST Miscellaneous Publication 286, Units of Weights and 

Measures. Price $2.50 SD Catalog No. C13 10286 Updated 6/17/98
iii

Acknowledgements

The author thanks the following for their input and contribution to the literature review: Nicholas 

Wilson, Curtis Urban, Charity Duran, Adam Klopp, Tony Sultana, Silvia Galvan-Nunez, Abe 

Meddah, and Brent Whitsitt.

Thanks to Andreas Haigermoser (Siemens AG), Raymond Lewis (Raymond Lewis, Ltd.), Eric 

Sherrock and Radim Bruzek (ENSCO, Inc.), Bernhard Metzger (Plasser American Corp.), 

Andreas Sinning (Trimble Railway GmbH), Davis Thompson and Steve Atherton (Balfour 

Beatty plc.), Arthur Clouse (Federal Railroad Administration), Aaron Brown and Stephen 

Ashmore (Union Pacific Railroad), Michael Craft (Amtrak), Felix Krupczynski (Holland LP), 

Edwin Reeves (Vista Instrumentation LLC), Jake Thomas (MERMEC Inc.), Yan Liu and Eric 

Magel (National Research Council Canada), and Brad Kerchof (formerly Norfolk Southern 

Corp.) for valuable discussions concerning their research, products, and/or track geometry 

measurement methods.

The team thanks the staff of the Library of the University of Illinois at Urbana-Champaign for 

their help with the literature review.
iv

Contents

Executive Summary ................................................................................................................... 1

1. Introduction ............................................................................................................. 3

1.1 Overview..................................................................................................................3

1.2 Objectives and Scope ...............................................................................................3

1.3 Sources.....................................................................................................................4

1.4 Organization of the Report .......................................................................................5

2. The Basics of Track Geometry ................................................................................ 6

2.1 Track Geometry Design Elements............................................................................6

2.2 Track Irregularities...................................................................................................9

2.3 Absolute vs. Relative Track Geometry; Distance and Wavelength Domains ........... 10

2.4 Space Curve ........................................................................................................... 13

2.5 Chordal (Versine) Measurements ........................................................................... 18

2.6 Additional Track Geometry Variables.................................................................... 19

2.7 Relationship between Space Curve and Chordal Offset........................................... 20

2.8 Effects of Track Irregularities on Rail Vehicles...................................................... 23

3. Overview of Process of Track Geometry Measurement and Characterization..........25

4. Types of Track Geometry Measurement Systems...................................................28

4.1 Relative and Absolute TGMS................................................................................. 28

4.2 Platform ................................................................................................................. 28

4.3 Principle of Operation ............................................................................................ 29

4.4 Autonomous TGMS ............................................................................................... 32

4.5 Vehicle Response Measurement ............................................................................. 33

5. Processing of TGMS Outputs.................................................................................35

5.1 Conversion of Absolute Track Geometry into Space Curve .................................... 35

5.2 Separation of Track Irregularities from Track Layout ............................................. 36

5.3 Restoration of Space Curve from Chordal Measurements....................................... 42

5.4 Verification of TGMS Outputs............................................................................... 43

6. Characterization of Track Geometry in the Distance Domain..................................45

6.1 Pre-Processing........................................................................................................ 45

6.2 Track Segmentation................................................................................................ 46

6.3 Exploratory Data Analysis...................................................................................... 47

6.4 Irregularity Analysis: Statistical Approach ............................................................. 48

6.5 Irregularity Analysis: Parametrization Methods...................................................... 56

6.6 Irregularity Analysis: Vehicle Response Methods ................................................... 57

6.7 Comparative Assessment of Distance Domain Track Geometry Characterization 

Methods ................................................................................................................. 58

7. Characterization of Track Geometry in the Wavelength Domain ............................63

7.1 Fourier Transform and Power Spectral Density Estimation..................................... 63

7.2 Cross Power Spectral Density, Coherence, and Transfer Functions......................... 66

7.3 Interpretation and Limitations of PSD .................................................................... 68
v 

7.4 Parametrization of PSDs......................................................................................... 69

7.5 Generalized Energy Index ...................................................................................... 71

8. Processing of Track Geometry Data for Vehicle Dynamic Simulations ...................72

8.1 Use of Measured Irregularities................................................................................ 72

8.2 Synthesis of Track Geometry Data ......................................................................... 74

9. Summary and Conclusions.....................................................................................76

10. Recommendations for Track Geometry Characterization Study ..............................78

11. References..............................................................................................................81

Appendix A. FRA Track Safety Limits (49 CFR §213) .............................................................91

Appendix B. Proposed Descriptions of Deterministic Track Anomalies (Hamid et al., 1983) ....93

Abbreviations and Acronyms....................................................................................................95
vi 

Illustrations

Figure 1. FRA DOTX 217 Track Geometry Car (FRA, 2019) .....................................................3

Figure 2. A simple curve .............................................................................................................6

Figure 3. MCO measurement with a 62-foot chord......................................................................7

Figure 4. Curves without (left) and with (right) transition curves.................................................8

Figure 5. Multicentered curves....................................................................................................8

Figure 6. Track gauge and superelevation....................................................................................8

Figure 7. Balance conditions.......................................................................................................9

Figure 8. An example of relative track geometry (i.e., profile, alignment, cross level, curvature, 

and gauge) displayed in a strip chart (FRA, 2017) ............................................................. 11

Figure 9. Track alignment data in distance and wavelength domain: space curve (bottom left), 

spectrogram (top left), PSD plot (top right) and normal probability plot (bottom right) – 

Data from DYNOTRAIN project (Haigermoser et al., 2013) ............................................. 12

Figure 10. The relationship between absolute track geometry and alignment............................. 13

Figure 11. Space curve corresponding to the track in Figure 10................................................. 14

Figure 12. Absolute and relative vertical track geometry ........................................................... 14

Figure 13. Non-tilting (left) and tilting (right) track-centered coordinate systems...................... 15

Figure 14. Space curve variable definitions............................................................................... 17

Figure 15. Versine (chordal) measurement ................................................................................ 18

Figure 16. Mid-chord offsets (2017 FRA Compliance Manual, Vol. 2, Ch.1, p.2.1.29).............. 18

Figure 17. Rail dip angle (left); switch entry (i.e., kink) angle (right)......................................... 20

Figure 18. A comparison between rail alignment in space curve and mid-chord offset format ... 20

Figure 19. Vertical track irregularity recorded in space curve and mid-chord offset formats...... 20

Figure 20. Magnitude of a space curve-to-versine transfer function vs. the ratio of irregularity 

wavelength to chord length; α=0.5 corresponds to a symmetric versine, also known as a 

mid-chord offset (Cohen and Hutchens, 1970; Ahmadian, 1999 and others) ...................... 21

Figure 21. Magnitude of a space curve-to-chordal offset transfer function vs. irregularity 

wavelength, calculated for standard chord lengths (31, 62, and 124 ft)............................... 21

Figure 22. Magnitude of theoretical vs. measured transfer function of MCO over space curve .. 22

Figure 23. FFT amplitude plot of space curve and mid-chord offset data ................................... 22

Figure 24. Vehicle suspension resonance modes........................................................................ 24

Figure 25. Example of a relationship between carbody resonant frequencies, vehicle speed, and 

track irregularity wavelength ............................................................................................. 24

Figure 26. Overview of track geometry measurement and characterization process................... 26
vii 

Figure 27. Manual and automated optical surveying.................................................................. 29

Figure 28. Traditional vehicle-mounted IMS and its schematic (Lewis, 2011)........................... 30

Figure 29. Vehicle-mounted chordal TGMS.............................................................................. 31

Figure 30. A simplified system for measurement of vertical irregularities (Lewis, 2011)........... 33

Figure 31. Curvature and alignment as functions of filter cutoff wavelength (data courtesy of 

Trimble GmbH)................................................................................................................. 38

Figure 32. Grade, superelevation, and profile as functions of filter cutoff wavelength (data 

courtesy of Trimble GmbH)............................................................................................... 39

Figure 33. Track centerline profile (top) and grade (bottom) as functions of filter cutoff 

wavelength ........................................................................................................................ 40

Figure 34. The relationship between wavelengths, vehicle speeds, and resonant frequencies; EN 

13848 wavelength ranges are highlighted (Adapted from (Lewis, 2011))........................... 42

Figure 35. Space curve restoration from MCO measurements ................................................... 42

Figure 36. Example of a cross-check of TGMS measurements: transfer function gain (top) and 

coherence function (bottom) (Lewis, 2011) ....................................................................... 43

Figure 37. Band-pass filters used to isolate cyclic profile irregularities associated with 

submultiples of rail length (Lewis, 2011)........................................................................... 46

Figure 38. PSD of unfiltered cross level signal (left) and PSD of cross level signal pre-filtered 

with a 200-meter high-pass filter (right), both calculated with Welch’s method (note the 

distorting effect of excessive number of segments) (Data from DYNOTRAIN project 

(Haigermoser et al., 2013))................................................................................................ 66

Figure 39. An example of coherence between left and right rail alignment (Hamid et al., 1983) 67

Figure 40. PSD of alignment, profile, and cross level, showing the effects of continuous 

stationary random process and a periodic deterministic process (Data from the 

DYNOTRAIN project (Haigermoser et al., 2013))............................................................. 68

Figure 41. Harmonic patterns as submultiples of rail length (Lewis, 2011)................................ 69

Figure 42. Proposed parametrization for PSD curves corresponding to different track classes 

(Hamid et al., 1983)........................................................................................................... 70

Figure 43. PSD of alignment, profile, and cross level irregularities from two European railways – 

dashed lines correspond to the theoretical PSD shapes described by the ERRI formulae 

(Data from DYNOTRAIN project (Haigermoser et al., 2013))........................................... 71

Figure 44. Basic layout of Minimally Compliant Analytical Track (MCAT) (49 CFR §213 

Appendix D)...................................................................................................................... 72
viii

Tables

Table 1. Carbody resonant frequencies...................................................................................... 23

Table 2. The relationship between space curve variables and filter cutoff wavelengths.............. 37

Table 3. Description of track anomalies (adapted from Hamid et al., 1983)............................... 57

Table 4. Alignment exception limits defined in United States Code of Federal Regulations (49 

CFR §213)......................................................................................................................... 91

Table 5. Profile exception limits defined in United States Code of Federal Regulations (49 CFR 

§213) ................................................................................................................................. 92

Table 6. Proposed formulae and parameter values for track anomalies (adapted from Hamid et 

al., 1983)........................................................................................................................... 93
1 

Executive Summary

From October 2018 to March 2019, the Federal Railroad Administration (FRA) sponsored 

Transportation Technology Center, Inc. (TTCI) to conduct a literature review on the methods of 

measurement and characterization of track geometry. The goal of the review was to summarize 

the current state of track geometry measurement and to provide recommendations on methods 

for processing and characterizing track geometry data collected under FRA’s Automated Track 

Inspection Program (ATIP).

The key findings of the literature review are summarized below.

1. Railroads around the world use many different methods to process and characterize track 

geometry data. Even within Western Europe, where operators generally follow 

Euronorms and Technical Specifications for Interoperability (TSI), there are significant 

differences between countries and rail operators.

2. Track geometry data can be represented in two broad forms: distance domain (i.e., using 

time histories of track irregularities as functions of the distance along the track) and 

frequency or wavelength domain (i.e., amplitude and phase or power spectral density of 

track irregularities as functions of signal wavelengths). Distance domain is the format 

used to assess track compliance with safety and quality norms. The majority of methods 

described in the literature are applicable to the distance domain.

3. Some flexibility is required for analyzing track geometry data from various track classes. 

In particular, filters with different cutoff frequencies may need to be used to process data 

from different classes of track.

4. Although experts have proposed or are using several alternative methods for track 

geometry characterization, the most common methods by far are those which extract 

simple statistical features from track geometry variables (e.g., alignment, cross level, 

etc.). These methods are maximum values and standard deviations. Railroad 

agencies/companies use maximum values to assess the safety of operations and use 

standard deviations to assess the overall track quality and to prioritize track maintenance.

5. Alternative methods of track geometry characterization are not conclusively proven to be 

superior to standard deviation-based statistics in terms of predicting the response of rail 

vehicles to track geometry. Different studies on the topic give conflicting or inconclusive 

results.

6. For track geometry analysis, track data should preferably be segmented based on track 

layout, special trackwork, and other features. For agencies using constant length 

segments, the most common length suggested is between 1/8 mile and 1/10 mile. 

7. Track irregularities generally should not be assumed to be described by a stationary, 

random process. Thus, standard deviation values do not provide a definite description of 

track geometry condition and should not be relied upon to predict the number or severity 

of track class defects, even though there is a correlation between class defects and 

standard deviation values.

8. Characterization of track geometry in wavelength (i.e., frequency) domain is widely used 

to generalize about overall characteristics of track classes, railroad systems, etc. The most 
2 

widely used method of wavelength characterization is power spectral density (PSD) 

functions calculated with different varieties of Fast Fourier Transform (FFT) algorithms.

9. Studies show strong coherence between left and right rail alignment, as well as between

left and right profile signals. However, mean alignment, mean profile, gauge, and cross 

level signals have low coherence (i.e., are relatively independent). 

Overall, the most widely used methods for track geometry characterization are refined versions 

of methods which were used in an FRA-sponsored study of track geometry in the 1970s-1980s. 
3 

1. Introduction

1.1 Overview

Track geometry is the position of rails in horizontal and vertical planes. It includes design 

features (e.g., tangents, curves, transition curves, superelevation) as well as irregularities (i.e., 

defects). 

Track geometry defects are the second-most common cause of derailments on Class I main line

track in the United States (Liu, 2012). Accurate and timely measurement of track geometry is 

critical for maintaining the safety of rail transportation, as well as for passenger comfort and the 

prevention of damage to rolling stock and lading.

Track geometry measurement has evolved from relying on manual and visual methods, to early 

track geometry cars equipped with analog instrumentation and strip chart recorders, and finally 

to modern Track Geometry Measurement Systems (TGMS), which record a wide range of 

parameters and use advanced digital instrumentation (Figure 1).

Figure 1. FRA DOTX 217 Track Geometry Car (FRA, 2019)

Processing and characterizing track geometry measurement involves a series of complex steps 

and a variety of methods, some of which are widely used and some of which are experimental. 

The selection of the method depends on the goal of the processing (e.g., planning for 

maintenance, preparation of data for computer simulations, etc.) and the type of data available.

From October 2018 to March 2019, the Federal Railroad Administration (FRA) sponsored 

Transportation Technology Center, Inc. (TTCI) to conduct a literature review on the methods of 

measurement and characterization of track geometry.

1.2 Objectives and Scope

This report reviews the current state of track geometry measurement and characterization

methods and provides an overview of:

• Basic definitions related to track geometry, including design elements and irregularities

• Causes of track irregularities and their effect on vehicle performance
4 

• Various forms of track geometry, including absolute track geometry, space curve, chordal 

offsets, power spectral density (PSD) functions, and mathematical methods of 

transitioning between these forms

• Modern methods of track geometry measurement, including surveying, inertial and 

chord-based systems, and vehicle response measurement methods

• Methods of characterizing track irregularities, such as track quality indices

• Process of preparing track geometry measurement for use with multi-body dynamics 

simulations

The report concludes with recommendations for processing the track geometry data collected 

under FRA’s Automated Track Inspection Program (ATIP).

1.3 Sources

The sources used in this literature review were identified using:

• The Transport Research International Documentation database by the Transportation 

Research Board of the United States National Academies of Sciences, Engineering, and 

Medicine

• Google Scholar search engine

• FRA’s eLibrary system

• Association of American Railroads (AAR) internal report database

• TTCI’s collection of technical literature

• Personal communication with researchers active in the field of track geometry and 

manufacturers and users of track geometry measurement systems

Many sources were identified through the literature reviews in these publications:

• (Haigermoser et al., 2015), (Karis, 2018), and (Lewis, 2011): comprehensive information 

on characterization of track geometry

• (Haigermoser et al., 2013): comparative assessment of track irregularity characterization 

methods 

• (Nielsen et al., 2013), (Moskal and Pastucha, 2016), and (Lewis, 2011): information 

about methods of measurement of track geometry

• (Berawi, 2013): characterization of irregularities in the wavelength domain

• (Sadeghi, 2010) and (Liu et al., 2015): track quality indices

Readers looking for additional information about specific topics are encouraged to review these 

papers.

The following sources contain additional information on topics that are closely related but 

outside the scope of this document:

• (Nielsen et al., 2013): short-wavelength irregularities, such as rail corrugation
5 

• (Berawi, 2013), (Soleimanmeigouni et al., 2018a and 2018b), (Higgins and Liu, 2018), 

(Elkhoury et al., 2018): track geometry degradation

• (Lindahl, 2001): design elements of track geometry

1.4 Organization of the Report

Section 2 describes the basics of track geometry, including terminology, track layout (i.e., 

design) elements, and different formats of recording track geometry data, including space curve, 

chordal offsets, and PSD data.

Section 3 provides a broad overview of the track geometry measurement and characterization 

process.

Section 4 describes the current state of TGMS.

Section 5 describes the initial steps in processing track geometry data, such as filtering and 

segmentation. 

Section 6 describes various methods used to characterize track irregularities in distance domain, 

including standard deviations and track quality indices.

Section 7 describes methods to characterize track geometry in wavelength domain, such as PSD 

functions. 

Section 8 outlines the steps necessary to prepare track geometry for use as input in multi-body 

dynamics simulations.

Section 9 summarizes the findings of the literature review.

Section 10 makes recommendations on the use of various characterization methods in the Track 

Geometry Characterization Study.


8 

Figure 4. Curves without (left) and with (right) transition curves

Multicentered curves (Figure 5) are adjacent simple curves with either no tangent section or a 

short section of tangent track separating them. Transition curves may or may not be present. 

Compound curves (i.e., adjacent curves with the same sign of curvature), reverse curves (i.e., 

opposite signs) and broken back curves (i.e., curves in the same sign with a short tangent 

separating them) are special cases of multicentered curves (Hay, 1982).

Figure 5. Multicentered curves

Vertical track geometry consists of constant grade sections and vertical curves, which are

usually designed as quadratic parabolas. Grade is the change in elevation (i.e., rise) over 

horizontal distance (i.e., run), often closely approximated as rise over distance traveled. Grade is 

usually expressed as a percentage (% or ‰). 

Track gauge (Figure 6) is the distance between the inner sides of rail heads measured at a 

certain height below the top of the rail (5/8-inch in North America, 14 mm in Europe). Standard 

gauge, which is used is most countries, is 56 1/2 inches, or 1435 mm. In tight curves, gauge may 

be intentionally made slightly wider than nominal to improve vehicle steering.

Figure 6. Track gauge and superelevation

10 

• Degradation of crossties and fasteners

• Poor weld geometry

• Rail surface wear

• Soil settlement

• Changes in ballast density and stiffness due to settlement, washout, scattering and 

tamping operations

• Vehicle-track interaction on small scale (e.g., rail corrugation, rail squats, etc.) and large 

scale (e.g., uneven ballast and soil settlement under vehicle loads); there is a significant 

positive feedback between increasing wheel-rail forces and degradation of track geometry 

(Elkhoury et al., 2018)

• Lateral track movement due to high lateral wheel forces and insufficient lateral stiffness 

of the track

• Track buckling due to thermal loads and insufficient restraint

The amplitude of track irregularities usually increases when they are measured under a vertical 

load (see Section 4.2).

The majority of modern TGMS sample data at intervals of about 1 ft, or 0.25 m, which means 

that even under ideal conditions they cannot measure irregularities with wavelengths less than 2 

ft. Furthermore, many TGMS apply band-pass filters with lower cutoff wavelength of about 9-13 

ft. Irregularities with shorter wavelengths, such as rail corrugation, require special equipment for 

measurement and are subject to standards and regulations that are separate from track geometry 

standards (Bracciali et al., 2009; Nielsen et al., 2013). These short irregularities are not covered 

in this report. 

2.3 Absolute vs. Relative Track Geometry; Distance and Wavelength Domains

Track geometry measurements are often separated into two broad categories: absolute and 

relative geometry.

Absolute (i.e., outer geometry) refers to the position of the track with respect to an absolute 

reference point. It allows establishing track location with respect to other structures, which is 

especially important for maintaining clearances around the track.

Absolute track geometry is measured using traditional or automated surveying methods and is 

recorded in a surveying coordinate system (i.e., northings, eastings, and elevation). The origin of 

the coordinate system can be at any point on or off the track; XY surface is parallel to the surface 

of the geoid, and Z axis is perpendicular to this surface (i.e., parallel to the direction of Earth’s 

gravitational force).

While absolute track geometry is important for track construction (and, in some cases, 

maintenance), it is not well-suited for evaluating the track’s compliance with track safety 

standards or for predicting rail vehicles’ responses to track irregularities.

Relative (i.e., inner geometry) is recorded in a track-centered coordinate system, with design 

geometry variables (i.e., curvature, superelevation, grade) and track irregularity variables (e.g., 

alignment, profile, gauge, cross level, etc.) being plotted as functions of distance along the track 
11 

(i.e., chainage). Such data is sometimes also described as being in distance domain and its 

graphical representation is referred to as a strip chart (Figure 8). Relative track geometry 

contains no information on the position or heading of the track with respect to absolute 

references, but it describes curves and perturbations in the track which affect vehicle behavior. 

Relative track geometry is what is being measured by most TGMS. It is well-suited for verifying 

track quality and compliance with track safety standards, as well as for predicting a vehicle’s 

response using multibody dynamics (MBD) simulation.

Figure 8. An example of relative track geometry (i.e., profile, alignment, cross level, 

curvature, and gauge) displayed in a strip chart (FRA, 2017)

Relative track geometry can be calculated from absolute track geometry easily and accurately 

(see Section 5.1). The reverse process can, in theory, be performed by integration, but the 

accuracy is low due to rapidly accumulating integration error, unless the relative track geometry 

measurements are combined with engineering survey measurements (Reedman, 2014).

Relative track geometry is recorded in one of two formats: the space curve format and the 

chordal offset (i.e., versine) format.

The space curve format is directly related to vehicle performance and is therefore used for MBD 

simulations and for assessment of track quality (Zhang et al., 2004; El-Sibaie and Zhang, 2004; 

Li et al., 2016). In many countries, track safety standards are based on space curve measurements 

(EN 13848-1,5,6). However, space curve is relatively complex to measure and process, and as a 

result, many agencies are either unfamiliar with it or avoid using it (Malone, 2007).

Chordal offset measurements do not directly relate to vehicle performance, although a recent 

large study showed that statistics based on chordal measurements perform similarly to statistics 
12 

based on space curve measurements in terms of predicting vehicle response to track geometry 

(see Section 6.7). Chordal measurements are intuitive and can be taken with simple hand tools. 

In many countries, including the United States, track safety standards are based on chordal 

measurements.

Space curve and chordal measurements are described in detail in Sections 2.4 and Section 2.5. 

Distance domain is useful for, among other things, examining individual track geometry defects 

and making safety-critical decisions, such as setting speed restrictions and prioritizing repairs. 

For this reason, most characterization methods and assessment criteria, such as track quality 

indices (TQIs), are designed for distance domain. These methods are described in Section 6. 

In wavelength domain, track geometry variables are presented in terms of amplitudes, phase, 

and/or signal power densities. These variables are plotted as functions of wavelengths or spatial 

frequencies (i.e., inverse wavelengths) of their signal. A plot of power spectral density vs. 

wavelength or spatial frequency is called a PSD plot, and a heat map of power spectral density 

as a function of wavelength and distance along the track is called a spectrogram (Figure 9).

Figure 9. Track alignment data in distance and wavelength domain: space curve (bottom 

left), spectrogram (top left), PSD plot (top right) and normal probability plot (bottom 

right) – Data from DYNOTRAIN project (Haigermoser et al., 2013)

Presenting track geometry data in the wavelength domain is useful for:

• Identifying periodical patterns in irregularities, such as cyclical dips associated with rail 

joints or welds

• Looking at the big picture and making general conclusions about the condition of a large 

section of track or even an entire rail system
13 

• Making qualitative comparisons between different sections of track, categories of track, 

or rail systems

• Describing the properties of filters and transfer functions

• Exploring relationships and correlations between different track geometry variables (e.g., 

alignment and gauge, gauge and cross level, etc.)

• Verifying measured track geometry and identifying problems in track geometry 

measurement systems

• Making general predictions about long-term vehicle behavior, for instance, by identifying 

cyclical track defects which can excite suspension oscillations (see Section 2.8) 

• Generating artificial stochastic track irregularities for computer simulations

Wavelength domain analysis, as well as methods of transitioning between distance and 

wavelength domain, are discussed in detail in Section 7. 

2.4 Space Curve

The principle behind the space curve is the separation of long length features of the track 

geometry, such as curvature and gradient, from short length features, such as alignment and 

profile deviations. This separation is achieved by defining a reference track trajectory. Space 

curve is a description of the deviations of measured track from the reference trajectory combined 

with the description of the gross geometry of the reference trajectory. The reference trajectory is 

not necessarily the same as design geometry (see Sections 5.1 – Section 5.2).

Figure 10 shows a plane view of a rail track. Solid lines represent the actual positions of the rails 

and of track centerline, dashed lines represent the reference trajectory of the track, and the 

distance between solid and dashed lines represents the space curve (shown in Figure 11). Note: 

chainage is defined as the measured distance along the track centerline. Typically, one chain is 

equal to 100 feet in length.

Figure 10. The relationship between absolute track geometry and alignment
14 

Figure 11. Space curve corresponding to the track in Figure 10

Track geometry in the vertical plane is described similarly (Figure 12). The geometry of the 

reference centerline is described in terms of grade or vertical curvature and the geometry of 

measured track is described in terms of profile and cross level (or, alternatively, in terms of left 

and right rail profiles).

Figure 12. Absolute and relative vertical track geometry

Space curve data is recorded in a track-centered coordinate system. There is no universally 

agreed-upon coordinate system, but one possible option is described below.

The origin of the system is located on the reference centerline trajectory. The X axis is in the 

direction of heading of the track, the Y axis points from the origin toward the nominal left rail 

position, and the Z axis points from the origin up. This coordinate system can be either tilting or 

non-tilting (Figure 13). In a non-tilting coordinate system, the Z axis always remains parallel to 

the Earth’s gravity. In a tilting coordinate system, the Z axis is always perpendicular to the plane 


17 

Figure 14. Space curve variable definitions
18 

2.5 Chordal (Versine) Measurements

In many countries, including United States (49 CFR §213 Subpart C) and Canada (TC E-54 

Subpart C), track safety standards are defined not in terms of space curve but rather in terms of 

chordal measurements, also called versines (Figure 15, Figure 16). Versine with ratio (α) of 0.5 

is known as a symmetric versine, also called a mid-chord offset (MCO).

The convenience of chordal measurements is that they can be taken easily in the field with 

simple hand tools.

Figure 15. Versine (chordal) measurement

 

Figure 16. Mid-chord offsets (2017 FRA Compliance Manual, Vol. 2, Ch.1, p.2.1.29)

Chordal offset is a function of both alignment irregularity (Figure 15) and track design curvature

(Figure 3). For example, a point in the body of a 10-degree curve has an MCO value of 10 inches 

for a 62-foot chord length if the rail at that point is perfectly aligned with its reference trajectory. 

For this reason, track geometry exceptions are defined not in terms of raw chordal values, but in 

terms of the deviation of chordal values from their localized averages. The FRA Compliance 

Manual specifies that for manual chordal measurements in curves, the averaging must be 

performed as follows:

• For Track Classes 1 through 5:

o Alignment (2017 FRA Compliance Manual Vol. II Ch. 1, §213.55):

 On tangent track: no averaging

 In a body of a curve: averaging is performed over measurements taken at 9 

or 17 overlapping uniformly spaced points over a total curve length of 248 

feet, regardless of chord length

20 

from a change in vertical rail profile gradient over a short wavelength. Dip angle is directly 

related to the vertical impact wheel loads and is sometimes used as a pre-indication of an 

impending rail end break. Due to the elasticity of rails, the magnitude of dip angle depends on 

vehicle speed, wheel load, and direction of travel. (Mandal et al., 2016; RAIB, 2014; prEN 

13848-1:2016). 

Kink angle, or entry angle, can be described as a lateral equivalent of a dip angle, although it 

occurs for very different reasons. Kink angle is a design variable of turnouts and is not 

commonly measured by TGMS.

Figure 17. Rail dip angle (left); switch entry (i.e., kink) angle (right)

2.7 Relationship between Space Curve and Chordal Offset

Qualitatively, the relationship between space curve and chordal data can be described as follows: 

if a reference trajectory is a straight line, then space curve is a series of mid-chord offset

measurement made with an infinitely long chord. For a curvilinear reference trajectory, the 

relationship is more complex (Figure 18).

Figure 18. A comparison between rail alignment in space curve and mid-chord offset 

format 

Figure 19 shows a dip in the track expressed in a space curve and chordal formats with different 

chord lengths. The distorting effect of chordal measurements can be seen clearly. 

Figure 19. Vertical track irregularity recorded in space curve and mid-chord offset formats

22 

with a track geometry car equipped with an inertial TGMS, which records track geometry data in 

MCO and space curve formats. Figure 22 shows the magnitude of three transfer functions for left 

rail alignment data:

1. Analytically calculated transfer function of a 62-ft MCO over space curve (same as the 

62-ft. MCO plot in Figure 21) 

2. Numerically calculated transfer function of synthetic 62-ft. MCO data over the space 

curve data recorded by the TGMS (Synthetic 62-ft. MCO data is calculated analytically 

from the space curve data recorded by the TGMS using the analytical equation for the 

transfer function) 

3. Numerically calculated transfer function of 62-ft MCO data recorded by the TGMS over 

space curve data recorded by the TGMS

Figure 22. Magnitude of theoretical vs. measured transfer function of MCO over space 

curve

Comparison of these transfer functions shows that due to the filter effects described earlier, 

MCO data recorded by TGMS contains more low frequency components than predicted by an 

analytical transfer function equation.

Occasionally an engineer is presented with a set of track geometry data collected with an 

unknown chord length, or even data whose type (space curve or MCO) is unknown. In these 

cases, frequency analysis is especially useful. Figure 23 shows the FFT amplitude plot of space 

curve and mid-chord offset data for left rail alignment measured with a track recording vehicle. 

Note the characteristic periodic peaks and valleys in the space curve data, which allow easy 

identification of chord length.

Figure 23. FFT amplitude plot of space curve and mid-chord offset data
23 

Regardless of chord length and methods for curvature removal, track geometry data recorded in 

the MCO format should not be used as an input for MBD simulations. At a minimum, it must 

first undergo a restoration (i.e., decoloring) process to convert it into space curve format (see 

Section 5.3).

2.8 Effects of Track Irregularities on Rail Vehicles

Gauge deviation. Narrow gauge can lead to excessive lateral wheel-rail forces and cause 

excessive wheel and rail wear. Wide gauge can decrease lateral stability of the vehicle and, in 

extreme cases, lead to wheel drop derailment (Sun et al., 2013; Wolf, 2015).

Cross level. Cross level irregularities cause wheel unloading on the affected side, which under 

extreme conditions may lead to derailment, especially when high lateral wheel forces are also 

present. In some cases, vehicle suspension can bottom out (i.e., its range of motion is exceeded), 

resulting in shock loads to the vehicle and damage to the vehicle, occupants, and cargo. In less 

extreme cases, it is a ride quality issue, inducing roll motion of the carbody. 

Warp and twist cause wheel unloading on the opposite corners of the truck (i.e., bogie) and of 

the vehicle, increasing the risk of flange climb derailment. The longer the wheel base and the less 

compliant the vehicle suspension, the more severe the wheel unloading due to track warp. This is 

one of the limiting factors for design length of transition curves.

Alignment irregularities cause excessive lateral wheel-rail forces, resulting in excessive wheel 

and rail profile wear and increasing the risk of flange climb derailment. They also may lead to 

excessive carbody lateral accelerations, resulting in poor ride quality. Out-of-phase alignment 

irregularities in left and right rail constitute gauge irregularities.

Profile (surface). When left and right rail profile irregularities occur in phase, they excite the 

pitch and bounce carbody modes (see explanation on cyclical irregularities below), resulting in 

cyclic wheel unloading, excessive vertical wheel-rail forces, and carbody accelerations. As with 

cross level defects, extreme cases may lead to the exceedance of suspension’s range of motion 

and damage to the vehicle. Out-of-phase profile irregularities are equivalent to cross level 

irregularities.

Cyclical irregularities and resonance. Even relatively small excursions of alignment, profile, 

or cross level can have catastrophic effects if multiple irregularities are spaced at such distances 

that the vehicles’ resonant frequencies are excited. As few as 2-3 irregularities, each of which 

below a safety threshold, may be enough to cause a derailment or damage track and vehicle 

components. 

Table 1. Carbody resonant frequencies

Carbody 

vibration 

mode

Typical frequency range (Hz) by vehicle type Track geometry 

excursion types likely 

to excite the mode

Locomotives Freight cars 

(empty)

Freight cars 

(loaded)

Passenger 

cars

Bounce 1.2–1.8 4.2–5.0 1.8–2.5 0.8–1.2 Profile

Pitch 2.0–2.7 5.0–7.5 2.7–4.5 0.8–1.3 Profile

Yaw 1.2–1.8 3.5–5.0 1.5–3.0 1.0–1.3 Alignment

Roll (lower 

center) 0.4–1.0 1.5–3.0 0.7–1.2 0.5–0.8 Cross level, Alignment

Roll (upper 

center) 2.2–2.6 5.0–6.0 2.5–3.8 1.1–2.0 Cross level, Alignment
24 

The body of a rail vehicle and its suspension elements can be modeled as a 3D mass-springdamper system with multiple modes of vibration. The modes that are of the most interest are 

bounce, pitch, yaw, roll (i.e., upper center), and roll (i.e., lower center). Resonant frequencies of 

rail vehicles vary but generally range between 0.4 Hz and 8 Hz (Table 1). Figure 24 shows the 

vehicle suspension resonance modes.

Excitation of any of these modes at a frequency close to its resonant frequency can have 

catastrophic consequences. 

Figure 24. Vehicle suspension resonance modes

Figure 25 shows an example of the relationships between resonant frequencies, vehicle speed, 

and wavelength for a hypothetical loaded hopper car. For example, vertical irregularities 

associated with rail length (39 ft) and its second harmonic (19.5 ft) may excite this railcar’s 

bounce frequency at speeds near 55 mph and 28 mph, respectively.

Figure 25. Example of a relationship between carbody resonant frequencies, vehicle speed, 

and track irregularity wavelength
25 

3. Overview of Process of Track Geometry Measurement and 

Characterization

Figure 26 shows an overview of track geometry measurement, processing, and characterization 

cycle based on a classification proposed by Haigermoser et al. (2015). White rectangles 

designate data, grey rectangles designate processes, and dashed shapes designate optional steps.

The cycle may vary considerably depending on the hardware used to measure track geometry, 

end goals, and desired characterization methods. The next sections of this report focus on 

specific steps on this cycle.

The cycle begins with one of the types of TGMS: inertial, chordal, or surveying (Section 4.3). 

Alternatively, track geometry may be assessed indirectly by measuring vehicle reactions (e.g., 

wheel-rail forces, car body accelerations, etc.) using a Vehicle Reaction Measurement System 

(VRMS). 

In the case of an inertial system, the output of TGMS sensors is processed (usually by software 

aboard a track geometry vehicle) and the data from various sensors is combined to yield a raw 

space curve, which contains variables such as gauge, alignment, curvature, cross level, and 

surface (Section 2.4). Depending on the wavelength measurement range of the system, this space 

curve data may already have design elements (i.e., track layout) separated from track 

irregularities; if this is not the case, it must be performed in a separate step.

In the case of a chordal measurement system, the measured track geometry is output in chordal 

offset format. In this case, too, track irregularities must be separated from track layout.

Output of surveying track geometry measurement systems usually is in a form of absolute track 

geometry (Section 2.3), which must be converted into space curve or chordal format before 

proceeding to next steps. This process is generally combined with the process of separating track 

layout from irregularities.

Track quality assessment methods for chordal and space curve are not interchangeable. 

Therefore, in many cases, it is necessary to convert chordal data into space curve or vice versa. 

Some data is lost during space curve to chordal conversion, and space curve usually cannot be 

fully restored from chordal measurements (Section 5.3).

Track irregularity data may undergo additional, optional steps before track irregularity 

characterization is performed. These steps may include filtering (for example, under EN 13848 

standards, track irregularities are separated into different wavelength ranges) and, in some cases, 

detection and extraction of track anomalies and special features (e.g., joints, turnouts, 

crossings).

Track characterization can be performed in the distance domain (i.e., space curve of chordal 

offset) or wavelength domain. Transition from the former to the latter is achieved by several 

different methods, most commonly by Fast Fourier Transform (FFT). 

27 

In the distance domain, the characterization steps are:

1. Segmentation of track into sections; consecutive analysis can be performed on each 

segment

2. Combining multiple track geometry variables into a single variable (e.g., average of 

left and right rail alignment)

3. Extraction of features (e.g., mean, median, maximum, and standard deviation) from 

individual and/or combined track geometry variables

4. Combining extracted features (e.g., sum of standard deviations of gauge and cross 

level) into a single variable; this is how most track quality indices (TQIs) are calculated

5. Performing statistical analysis of features (e.g., calculating the distribution of track 

quality indices amongst track segments) 

All the steps except Step 3 (extraction of features) are optional.

If track irregularity data is transformed into wavelength domain via parametric or nonparametric methods, the resulting spectral density functions can be described analytically.

Collected and processed track geometry data can be used to predict rail vehicles’ response to 

track. It may be done directly via vehicle response analysis (VRA) methods such as pointmass-acceleration (PMA), neural networks (NN), or, after additional processing, via MBD 

simulations. As an alternative to using measured data for simulation, synthetic track geometry 

data can be created by combining synthetic track layout with stochastic and/or deterministic 

track irregularities.
28 

4. Types of Track Geometry Measurement Systems

Below is a brief overview of the most common types of track geometry measurement systems. 

This section only discusses TGMS which collect comprehensive track geometry data, including 

curvature, grade, cross level, gauge, alignment, and profile. Simpler systems which only measure 

some of the track geometry variables (most often gauge and cross level) are outside of the scope 

of this section.

4.1 Relative and Absolute TGMS

Relative TGMS, which typically use either inertia-based or chordal methods, constitute the 

majority of both trolley-based and vehicle-mounted systems. Compared to absolute TGMS of 

similar accuracy and portability, they tend to be more affordable and simpler to use.

The majority of absolute TGMS rely on optical surveying methods (see Section 4.3). Some of 

the new TGMS avoid the limitations of optical surveying by using combinations of inertial 

measurement systems (IMS), high-accuracy global navigation satellite systems (GNSS), and/or 

machine vision (Engstrand, 2011; Pinter, 2012; Chen et al., 2015; Chen et al., 2018; Trimble, 

2017). Almost all absolute TGMS systems are mounted on small trolleys, although track 

recording vehicles with absolute TGMS also exist (Vogelaar, 2017).

When users want to only measure relative track geometry, they may still benefit from using an 

absolute TGMS, which provide complete control over conversion of absolute track geometry 

into relative track geometry data, allowing the use of linear regression and curve fitting, 

customization of filters, etc.

4.2 Platform

TGMS can be mounted on hand-pushed trolleys, self-propelled carts, road-rail (i.e., hi-rail) 

vehicles, dedicated track geometry vehicles, and revenue service vehicles. Aside from the 

obvious logistical and financial considerations, it is important to recognize that track geometry 

measurements are affected by the stiffness of the track and the weight of the measuring vehicle. 

The relationship between wheel load and track geometry irregularities is not straightforward. It 

depends on the condition of ballast and soil, type of track structure, type of rail fasteners, etc. 

Furthermore, the extent to which a truck deflects rails (and therefore measured amplitudes of 

gauge and alignment irregularities) depends in part on its dynamic performance, such as curving 

characteristics.

In the United States, regulations require that in certain cases “track geometry measurements shall 

be taken no more than 3 feet away from the contact point of wheels carrying a vertical load of no 

less than 10 kips per wheel, unless otherwise approved by FRA” (49 CFR §213.333(b)).

If track geometry data is being used to evaluate track condition, then ideally it should be 

measured with a track geometry measurement car with an axle load representative of other 

vehicles on that line. If measured track geometry data collected in a loaded condition (i.e. loaded 

track geometry from a track geometry car) is being used as input (i.e. unloaded geometry) for 

MBD simulations, then using actual track stiffness parameters in the simulation may result in 

simulated loaded track geometry and wheel-rail forces that are somewhat different than reality 

(Section 8.1).
29 

4.3 Principle of Operation

Manual measurements are simple in principle but time-consuming and labor-intensive. 

Manual chord surveying measures relative track geometry in chordal offset format. It is 

performed using a chord of defined length and a ruler (Figure 15, Figure 16). This method often 

is used to verify track geometry defects measured by vehicle-mounted systems.

Manual optical surveying measures absolute track geometry. It is performed using a theodolite

or an automated geodetic total station (Figure 27, top). Vertical and lateral coordinates of a point 

on one of the rails with respect to a global coordinate system are measured; vertical and lateral 

coordinates of a point on the opposing rail are estimated from the coordinates of the original rail, 

combined with gauge and cross level at that point, measured with a gauge bar (Stow and 

Andersson, 2006).

Figure 27. Manual and automated optical surveying

Automated optical surveying is performed using track surveying trolleys. A typical setup 

consists of a wayside total station and a reflective target mounted on a trolley, which also 

contains sensors for cross level and gauge measurement. Alternatively, a total station can be 

mounted on a trolley and use wayside reflectors (Figure 27).

The trolley can make measurements in either kinematic mode (i.e., at walking speed) or in a 

more accurate stop-and-go mode. The main disadvantages of these systems are their slow speed 

of measurement and short range (tens or hundreds of meters), which is further limited by 

topography and atmospheric conditions (higher air temperature and larger wind speeds decrease 

the useful range). Range and accuracy can be increased by relocating a total station and making 

sequential overlapping measurements, but this process is relatively cumbersome and timeconsuming, limiting most absolute TGMS systems to use on trolleys intended for measuring 
30 

short sections of track. Some optical surveying systems combine surveying data with 

measurements from inertial measurement systems, as described below.

Inertial measurement systems (IMS) can be mounted on either the carbody or truck frame of a 

track recording vehicle, or, less often, on a portable trolley (Chen et al., 2015; Sundaram and 

Wilson, 2016; Trimble, 2017).

They combine the data from multiple sensors:

• An odometer to measure chainage

• Non-contact (usually optical) or contact sensors which determine the relative position of 

rails with respect to the sensor

• A combination of accelerometers and gyroscopes, which determines the relative 

acceleration and velocity of the measuring device with respect to ground and integrates 

these data to calculate the relative position

Traditional vehicle-mounted IMS have accelerometers and gyroscopes mounted on the truck 

frame or carbody, and displacement sensors (such as linear variable differential transformers 

(LVDTs)) mounted across vehicle suspension elements (Figure 28). Such systems are custom 

designed for each vehicle. On the other hand, many modern IMS are self-contained units 

mounted entirely to the carbody or truck frame and do not measure displacement across 

suspension. Each approach has advantages and disadvantages. Mounting sensors to an unsprung 

mass, such as an axle box, creates a more rigid coupling between sensor readings and track 

irregularities, but it places the sensors in a harsh vibration environment and requires measuring a 

very wide range of frequencies. TGMS with optical sensors mounted to the carbody may be 

limited to the maximum curvature of the track they are able to measure due to lateral offsets of 

the mounting location on the car body relative to the rails. Ultimately, there are systems of either 

type on the market that are capable of measuring space curve to EN 13848 specifications.

 

Figure 28. Traditional vehicle-mounted IMS and its schematic (Lewis, 2011)

One notable limitation of IMS is a minimum vehicle speed required for measurement. The lower 

the speed of the vehicle, the lower the magnitudes of the outputs of accelerometers and 

gyroscopic sensors, and below a certain speed signal to noise ratio it becomes too low for 

accurate measurements (Lewis, 2011). Therefore, the lower the speed of the vehicle, the shorter 

the maximum wavelength of a track irregularity that the IMS can measure accurately; 

measurement of long wavelengths usually requires much higher speeds than shorter wavelengths 
31 

(see discussion of wavelength domains in Section 5.2). However, due to recent improvements in 

sensor design and data processing, some of the newer IMS are capable of measuring track 

geometry at low speeds (i.e., ~5 mph) and/or can accommodate limited-duration stops. Such 

systems are often described as having “zero-speed” or “extended speed range” measurement 

capabilities.

Chordal systems are automated versions of manual chord survey (Figure 16). They measure 

lateral and vertical distances between each rail and the frame of the track recording vehicle or 

trolley at a minimum of three points (Figure 29). Where required, corrections can be made for 

the distortion resulting from the bending of the vehicle frame (Haigermoser et al., 2015). 

Resulting chordal measurements can be converted into space curve using a restoration process 

(Section 5.3). Asymmetric chordal systems have fewer zeros (i.e., wavelengths with zero gains 

that are impossible to restore) but introduce a phase distortion. Even for asymmetric systems, the 

larger the ratio of track defect wavelength to the chord length, the more difficult it is to 

accurately restore that wavelength due to decreasing signal/noise ratio (Figure 20). For this 

reason, small trolley-based chord systems are limited to short wavelengths, making them 

unsuitable for measuring high-speed rail tracks. Vehicle-mounted chord-based systems can 

measure longer wavelengths.

Figure 29. Vehicle-mounted chordal TGMS

Despite these limitations, chordal systems have several advantages over inertial systems. Chordal 

systems tend to be mechanically simpler and less costly than IMS. Their accuracy does not 

depend on vehicle speed and they can make static measurements. Nevertheless, because of the 

ongoing improvements to the design of IMS, vehicle-mounted chordal systems are becoming 

increasingly rare. Trolley-mounted chordal systems are still used widely.

Some chordal systems combine chordal measurements with other types of measurements to 

overcome the typical disadvantages of chordal systems. One of trolley-mounted systems uses a 

principle called “differential difference method,” which combines versine and slope 

measurements (Naganuma and Yada, 2016) and other systems combine chordal and inertial 

measurements (Yazawa and Takeshita, 2002; Yada et al., 2017).

With proper data processing, both chordal and inertial systems can produce relative track 

geometry measurements of acceptable quality, but the limitations of each must be understood.

Global Navigation Satellite Systems/Global Positioning Systems (GNSS/GPS) by themselves 

do not have sufficient accuracy to be used for track geometry measurements, because even highaccuracy differential GNSS used in this application have an accuracy of 0.5-1.0 inch (Szwilski et 

al., 2003). Generally, GNSS mounted on a track recording vehicle are used to record the location 

of the track geometry defects, but not to measure their magnitude. 

 α = a/(a+b)

versine = h2–(h1+ α (h3–h1))
32 

However, some recently designed TGMS blend the data from high-accuracy GNSS with inertial 

or chordal systems, which allows measurement of both absolute and relative track geometry 

while overcoming some of the problems of optical surveying systems (Luck et al., 2001; Kreye 

et al., 2004; Chen et al., 2015; Trimble, Inc., 2017). Other systems combine data from GNSS and 

optical surveying systems (Mahalakshmi and Joseph, 2013; Jiang et al., 2017).

The use of Doppler Light Detection and Ranging (LIDAR) systems to measure track chainage 

and curvature has been investigated. Their advantages over traditional methods of measurement 

of chainage (i.e., tachometers and encoders) and curvature (i.e., gyroscopes) include accuracy at 

low speeds and not being susceptible to wheel slip and wheel tread wear (Wrobel, 2013; Andani, 

2016; Andani et al., 2018).

Additionally, more complex LIDAR-based machine vision systems have recently emerged on 

the TGMS market. They can create 3D point clouds, allowing the measurement of not only track 

geometry (absolute and relative) but also clearances between track and nearby structures. 

(Vogelaar, 2017; Burton, 2018). Non-LIDAR machine vision systems also are being developed 

(Gabara and Sawicki, 2018).

Position determination: One of the challenges in the design and operation of TGMS is 

determining the accurate absolute position of track geometry defects (i.e., milepost). Odometers 

often have insufficient accuracy and GNSS does not work in tunnels; furthermore, GNSS is not 

sufficiently accurate to identify the track number in multi-track territory. This may necessitate 

more complicated methods, such as installing radio-frequency identification (RFID) tags or 

special processing of accelerometer data (Broquetas et al., 2012).

4.4 Autonomous TGMS

In the last decade, advances in electronics and software development led to a development of 

autonomous track geometry measurement systems (ATGMS). These systems are mounted on 

revenue service trains and allowed to function unattended, which allows more frequent and less 

costly inspections than traditional manned TGMS. Collected track geometry data is transmitted 

wirelessly to a remote operator (Morant, 2016; Stuart, 2017; Higgins and Liu, 2018). Railroads, 

transit agencies, and regulatory bodies show great interest in ATGMS and have been conducting 

extensive tests. However, at this time, their widespread implementation is being impeded by 

issues (Morell, 2017) such as:

• Difficulties with precise location of the defects, especially distinguishing between tracks 

in multi-track territory

• Challenges with assuring data quality and preventing false positives without personnel 

examining data in real time (see Section 6.1) 

• Logistical difficulties (procedures for hardware maintenance and data handling are not 

well established)

• Regulatory issues (unclear requirements as to remediation of defects revealed during 

automated inspections)

Additional technical challenges may arise depending on the specifics of vehicles on which 

ATGMS are installed. For example, placing an ATGMS on a freight railcar may subject it to a 

harsh load environment with high accelerations and natural frequencies that vary with loading 
33 

condition; the vehicle’s weight and curving characteristics can cause ATGMS measurements to 

differ from those taken by a traditional track geometry vehicle on the same route (see Section 4.2).

It is expected that within the next few years use of ATGMS will become much more widespread.

4.5 Vehicle Response Measurement

The end goal of measuring track geometry is to ensure vehicle safety and ride comfort. 

Therefore, it is often beneficial to measure vehicle performance directly by recording wheel-rail 

forces and/or carbody and truck accelerations, and to make conclusions about track condition 

from these measurements.

Vehicle response measurement systems (VRMS) supplement traditional track geometry 

measuring systems, but do not replace them for these reasons:

• Vehicle response does not clearly differentiate between irregularity types or measures their 

magnitudes. 

• Response is vehicle-specific. Lack of response by one vehicle to a track defect does not 

mean it is safe for other vehicle types or even for the same vehicle at a different speed; 

correlation is not straightforward.

Direct measurement of wheel-rail forces requires use of instrumented wheelsets, which are 

expensive and labor-intensive to design, use, and maintain. Accelerometers do not have those 

problems, but their readings do not directly describe wheel-rail forces.

In the United States, vehicle response measurement is mandated as a part of vehicle/track system 

qualification for Class 6 track speeds and above, or for any curving speed producing cant 

deficiency of more than 5 inch (49 CFR §213.333, §213.345). An overview of American and 

International Vehicle-Track Interaction (VTI) safety standards is given in Marquis et al. (2014). 

Vehicle response measurements can be used to calculate vertical track irregularities. 

Accelerometers are mounted onto axle boxes or carbodies, and the resulting accelerations are 

filtered and integrated to calculate vertical space curve, acting as a simplified IMS (Figure 30). 

Figure 30. A simplified system for measurement of vertical irregularities (Lewis, 2011)

Accurate calculation of lateral track irregularities (i.e., alignment) with such methods is not 

feasible, because wheelset and track are relatively weakly coupled in the lateral direction due to the 

clearance between the wheel flanges and rails. Weston et al. (2015) review the methods of track 
34 

geometry measurements from in-service vehicles; Al-Nazer (2014) describes FRA research on this 

topic; Karis (2018) gives an overview of methods used on Japanese high-speed railways.



38 

specifies upper and lower boundaries for filter transfer function and recommends FIR filter for 

real-time processing and FIR or IIR (second order Butterworth) filters for post-processing. 

Filter cutoff wavelength. The choice of a cutoff wavelength for a filter is far from 

straightforward. Recall that the main goal of filtering is to separate track layout from track 

irregularities. The difficulty is twofold. 

1. Some track irregularities can be longer than some track design elements. Therefore, a 

shorter cutoff wavelength may filter out some track irregularities, while a longer cutoff 

will make track design elements look like track irregularities.

2. Track design elements usually do not have a sinusoidal shape. In fact, design curvature 

and superelevation are usually not smooth functions, causing the transitions between 

design elements to seem like irregularities.

This problem is most severe in turnouts and in curves which have no spirals or short spirals. In 

these cases, excessively long cutoff wavelengths will result in transitions between curves and 

tangents being interpreted as large alignment deviations. If measured track geometry data shows 

excessively large alignment deviations near the beginning and the ends of the curves, but not in 

the body of the curve or on tangent track, filter cutoff wavelength may be too small for the short 

transition curves in the particular track. In addition, a turnout’s entry angle may manifest itself as 

a cusp-shaped alignment anomaly.

Figure 31 and Figure 32 show the space curve calculated from absolute track geometry 

measurements made by a track surveying trolley. 

Figure 31. Curvature and alignment as functions of filter cutoff wavelength (data courtesy 

of Trimble GmbH)
39 

Figure 32. Grade, superelevation, and profile as functions of filter cutoff wavelength (data 

courtesy of Trimble GmbH)

Note the effect of filter parameters on the shape of the space curve. Longer cutoff wavelength 

results in smoother curvature, superelevation, and grade plots, but larger amplitudes of alignment 

and profile deviations. For these reasons, when space curve data is used to evaluate a track’s 

compliance with safety standards or for calculations of TQI, the filter characteristics should be 

specified in detail.
40 

Figure 33 illustrates the same concept. It shows the space curve corresponding to the dip in a 

track measured by a track surveying trolley and processed with the linear regression method and 

with different filters. The apparent depth of the dip changes significantly depending on the 

processing method.

Figure 33. Track centerline profile (top) and grade (bottom) as functions of filter cutoff 

wavelength 

Another issue to consider is the operating speed limit on the track and the expected vehicle 

response at that speed. For example, on an FRA Class 1 track (speed of 15 mph for passenger 

vehicles and 10 mph for freight vehicles), it may not be practical to measure 300 ft (100 m) track 

irregularities, because their effect on vehicles at Class 1 track speeds would be insignificant. 

Furthermore, most inertial TGMS systems would not even be able to repeatedly measure such 

long wavelength irregularities at such low speeds. At the same time, Class 6 track is unlikely to 

have sharp curves and short spirals, so a longer cutoff wavelength is appropriate. 

In the United States there is currently no regulation or standard for the filter cutoff length for 

space curve, and the decision is left to the individual railroads and TGMS manufacturers. For 

high-speed operations in the Northeast Corridor, Amtrak uses upper a cutoff wavelength of 400 

ft (122 m), noting these larger wavelengths are mainly measured for ride quality rather that 

safety reasons (Li et al., 2016); North American freight and transit operators who choose to 

measure space curve generally use an upper cutoff wavelength between 70 to 100 ft (20-30 m), 

and FRA ATIP track geometry vehicles use an upper cutoff wavelength of 400 ft. European 
41 

standards and regulations use a tiered approach. Euronorm EN 13848 specifies four wavelength 

domains: 

• D1 domain, which includes alignment and profile wavelengths between 3 and 25 m (10 

and 82 ft); the lower cutoff can be adjusted down to 1 m (3 ft.). Irregularities in D1 

domain are measured for all track classes.

• D2 domain, which includes alignment and profile wavelengths between 25 and 70 m (82 

and 230 ft). D2 domain irregularities are measured for track with speed limit >160 km/h 

(99 mph). 

• D3 domain, which includes alignment wavelengths between 70 and 200 m (230 and 656 

ft) and profile wavelengths between 70 and 150 m (230 and 492 ft). D3 domain 

irregularities are measured for track with speed limit >250 km/h (155 mph).

• The optional D0 domain, described in the recent draft of Euronorm prEN 13848-1:2016, 

includes wavelengths between 1 and 5 m (3 and 16 ft). Other than the minimum sampling 

distance (0.1 m, or 0.3 ft), no requirements for this domain are currently defined.

This approach offers multiple benefits:

• It offers some flexibility in separating track design elements from track irregularities. As 

previously noted, rail tracks with higher speed limits tend to have shallower curves and 

longer spirals, allowing for longer cutoff wavelengths.

• Short-wavelength irregularities (i.e., D1 domain) and long-wavelength irregularities (i.e., 

D2 and D3 domains) can be assessed separately using different acceptance criteria. This 

is beneficial because long irregularities tend to have larger amplitudes and would 

dominate space curve if they were lumped together with short irregularities, and because 

shorter defects usually produce more severe vehicle reactions for the same amplitudes 

(Haigermoser et al., 2015).

• As vehicle speed increases, it becomes more sensitive to long irregularities, as can be 

illustrated by speed-wavelength-frequency relationship: 

A similar tiered approach is used in the United Kingdom, but the cutoff wavelength lengths 

between the tiers of irregularities are different from EN 13848 (Lewis, 2011).

Figure 34 illustrates the relationship between track irregularity wavelengths, vehicle speed, and 

resonant frequencies that are typical for rail vehicle suspension. The relationship suggests that 

the EN 13848 wavelength ranges may not be wide enough to capture irregularities that may 

excite some of the lower resonant frequencies (i.e., carbody roll); however, roll mode is mainly 

excited by cross level irregularities, which are not supposed to be filtered.
42 

Figure 34. The relationship between wavelengths, vehicle speeds, and resonant frequencies; 

EN 13848 wavelength ranges are highlighted (Adapted from (Lewis, 2011)) 

For a detailed discussion of wavelength range as function of vehicle type, speed, and other 

considerations, see (Lewis, 2011).

5.3 Restoration of Space Curve from Chordal Measurements

Calculating chordal offset from measured space curve is simple; the inverse process (i.e., 

decoloring) is far more complicated. It is possible to partially restore space curve from chordal 

offset measurements (Cohen and Hutchens, 1970; Mauer, 1995; Grassie, 1996; Aknin and 

Chollet, 1999; Chung and Ham, 2004; Glaus, 2006; Naganuma and Yada, 2016; Wang et al., 

2018a), or to extrapolate long chord offset data from shorter chord lengths (Carr et al., 2002), but 

a complete restoration of space curve data from MCO measurements alone is impossible. Track 

defects whose wavelength are close to even submultiples of chord length are irreversibly lost, 

and restoring perturbations with large wavelengths is difficult due to low signal to noise ratio. 

The process is further complicated by the effects of filters used to average MCO values over a 

distance. 

Figure 35 shows a FFT magnitude illustrating an attempt to restore a space curve from MCO 

data. The first data series is the space curve measured by the TGMS aboard a vehicle; the second 

data series is a 62 ft MCO data recorded by the same TGMS; the third data series represents the 

space curve restored from the chordal measurements.

Figure 35. Space curve restoration from MCO measurements
43 

In this example, track geometry data is restored remarkably well for wavelengths below ~85 feet 

(which is the cutoff wavelength used by this TGMS for computing the space curve). However:

1. Inevitably, there are holes in the restored space curve corresponding to the submultiples of 

chord length.

2. The restored space curve overestimates the amplitudes of long-wavelength perturbations 

(see Figure 22 and the accompanying discussion).

The quality of restoration may be improved by making asymmetric chordal measurements 

(Figure 20), or by combining chordal measurements with different chord lengths. With any 

restoration method, users must understand the range of wavelengths they are interested in 

measuring and the limitations of the restorative method. Frequency analysis can be a valuable 

tool in evaluating the quality of restoration (Haigermoser et al., 2013; Grabner, 2013a and 

2013b). 

5.4 Verification of TGMS Outputs

As discussed earlier, TGMS generally outputs alignments and profiles for left and right rails 

separately instead of centerline alignment and profile. Since TGMS also measure gauge and 

cross level, there is some redundancy between different measurement channels (see Table 2 and 

Equations (9)-(10)). If cross level and profile or gauge and alignment are measured with different 

sensors, this redundancy can be used to cross check the accuracy of data collected by the TGMS 

(see EN 13848-2:2006, Section A.3.2). This cross-check can be done in distance domain or 

wavelength domain by calculating cross-power spectral density, coherence, or transfer function 

(see Figure 36 and Section 7.2). Note that not only the magnitude and power but also the phase 

relationship between two outputs is important. Therefore, coherence or cross-power spectral 

density by itself is not a sufficient check.

Figure 36. Example of a cross-check of TGMS measurements: transfer function gain (top) 

and coherence function (bottom) (Lewis, 2011)
44 

Duplicate data in various channels can be a concern if the track geometry data is being collected 

for MBD simulation. For instance, a lateral track defect of a certain wavelength may be recorded 

in both curvature and alignment channels, which will cause the simulated vehicle’s response to 

be overestimated. To avoid this, users of track recording vehicles must have a solid 

understanding of the given TGMS and its operating settings. For example, the software aboard a 

track recording vehicle may allow the operator to change the cutoff wavelength for alignment, 

but the measured track curvature may retain the same frequency content, leading to either 

redundant or missing alignment data. Choice of cutoff wavelength and methods of averaging 

MCO values have additional important implications (see Section 5.2).
45 

6. Characterization of Track Geometry in the Distance Domain

In most cases, track geometry is assessed in the distance domain.

During this process, the outputs of a TGMS are being pre-processed and usually divided into 

shorter track segments. These segments are then analyzed using statistical, parametrization, or 

vehicle response-based methods. 

With statistical methods, simple mathematic manipulations may need to be performed first (for 

example, track centerline alignment may be calculated from left and right rail alignment). For 

each track segment, statistical features are calculated. These features fall into one of two broad 

categories:

1. Features characterizing isolated defects, such as minima, maxima, and peak-to-peak 

values of track geometry variables (e.g., alignment, cross level, etc.), are used to 

characterize isolated defects, identify safety-critical defects, help determine maximum 

safe speed, and identify a need for immediate repairs.

2. Features characterizing the overall track quality of a segment, such as standard 

deviation of alignment and profile, are used to characterize the overall track roughness as 

a proxy for ride quality, or to prioritize track maintenance. Sometimes multiple statistical 

features are combined into parameters (i.e., TQIs).

The division between the two types of features is often blurred. The number and severity of 

isolated defects tends to correlate with overall track quality (Scanlan et al., 2016; Haigermoser et 

al., 2013), although this correlation is far from perfect. Furthermore, TQIs may be based on 

counting the number of defects in a track segment.

Parametrization methods involve describing individual track irregularities using deterministic 

functions, wavelets, and triangles (Section 6.5).

Vehicle response-based methods of various complexity attempt to describe track irregularities 

in terms of predicted vehicle reaction forces (Section 6.6).

Finally, features from many track segments can be aggregated and analyzed; for example, 

probability distribution of standard deviations among track segments may be calculated.

6.1 Pre-Processing

Data cleanup. Raw track geometry measurements often contain artifacts and other invalid data 

which must be removed or overwritten before further processing. This data includes:

• Segments of data where alignment and profile measurements are unreliable because the 

speed of the track geometry vehicle was too low for an accurate measurement

• Dropouts and spikes in signal due to:

o Bolted joints and special track work, such as guardrails, frogs, and diamond 

crossings, where laser sensors fail to properly measure rail position

o Obstruction of sensors by dust, debris, sun glare, rain, snow, etc.

o Other malfunctions 
46 

Removal of signal artifacts and other invalid data is a challenging process. Ideally, an observer 

should be present aboard a track geometry measuring vehicle to identify and mark locations with 

unreliable measurements. When this information is unavailable, some artifacts can be detected 

and removed automatically in post-processing, e.g., by setting a threshold for rate of change of 

signal (Jia et al., 2014). For example, a 2-inch increase in gauge over a 1-ft distance is likely an 

artifact and not a real defect, since the track geometry vehicle did not derail or was otherwise 

damaged.

Position synchronization. A challenge arises when track geometry data from two or more 

measurement runs needs to be overlaid to assess repeatability of measurements or degradation of 

track geometry over time. Imperfections in measuring distance along the track (e.g., wheel 

tachometer error, lack of GNSS data) make this process quite complicated. Several solutions to 

this problem have been proposed (Hanreich et al., 2002; Jia et al., 2014; Xu et al., 2014; Wang et 

al., 2018b; Weston et al., 2015). A closely related problem is matching the location of defects to 

the correct milepost locations (Xu et al., 2016; Pedanekar, 2006).

Filtering. Pre-processing of track irregularity data may require additional filtering. For example, 

EN 13848 specifies different acceptance criteria for different irregularity wavelength ranges 

(e.g., D1, D2, D3). If the space curve has not already been broken into these ranges by the 

software aboard the TGMS, this step must be performed separately. Some railroads and 

regulatory bodies apply band-pass filters designed to detect certain irregularities which are likely 

to cause vehicle resonance (Figure 37).

Figure 37. Band-pass filters used to isolate cyclic profile irregularities associated with 

submultiples of rail length (Lewis, 2011)

6.2 Track Segmentation

It is often necessary to break down data from long track sections into shorter sections to 

characterize each section separately. The sections can be of fixed length. EN 13848-5:2017 gives 

a typical value of 200 m (656 ft) for calculating standard deviation; most European railway 

operators use this interval (Haigermoser et al., 2013). In Japan, track quality index is calculated 

over 100 m (329 ft) and 500 m (1,640 ft) segments (Liu et al., 2015). In China, 200 m and 500 m 

segments are used (the latter is used for high-speed track). The 200 m length was proposed for 

calculating TQI in the Beijing Metro system (Liu et al., 2015). Sadeghi et al. (2017) used 1 km 

(3,280 ft) segment lengths. In Sweden, Q-Value is calculated over 1 km (3,280 ft) (Berawi, 

Cyclic 4.5, 6, 9, 12.8 

1

10

100

1.0 10.0 Wavelength metres 100.0

Response % 

4.5 6 9 12.8











58 

specific analysis (e.g., MBD simulations or vehicle response measurements). These methods use 

track geometry measurements as input and attempt to predict the vehicle’s response using 

analytical or empirical functions (Haigermoser et al., 2015). The complexity of these methods 

varies widely. For example, a series of MBD simulations may be performed in advance on a 

model of a specific vehicle, and the results are then used to derive a linear system of equations 

that ties track geometry variables to vehicle response. Features extracted from measured track 

geometry data can then be input into this linear system to yield outputs describing the vehicle’s 

response to the measured track geometry.

EN 13848-6 VRA Method. The vehicle response analysis method described in EN 13848-

6:2014 relies on detailed and well-validated MBD models of a vehicle to simulate the vehicle’s 

response to a large variety of sinusoidal-shaped track irregularities of various amplitudes and 

wavelengths. Simulations are conducted at various speeds, track curvature, and superelevation 

values. The minimum and maximum lateral and vertical wheel forces and body accelerations are 

extracted from simulation results and normalized with respect to the safety limits in EN 14363. 

These vehicle reaction statistics are then related to the amplitudes and mean gradients of the 

track irregularities via a set of linear equations, which are solved to obtain a series of 

coefficients. To analyze measured track geometry, the space curve is broken into a series of 

individual irregularities. Amplitudes and gradients of these irregularities are combined with the 

coefficients calculated from MBD simulations to yield predicted vehicle reactions.

Neural Networks. Neural network (NN) approaches have been developed by the ERRI (ERRI, 

1999, as cited in Haigermoser et al., 2015) and TTCI (Li et al., 2006). These approaches segment 

the track into sections, extract track geometry features from each section, and relate them to 

vehicle response parameters, such as percentiles of vertical and lateral wheel forces, via neural 

networks. The weights in the NN, which describe the relationships between inputs and outputs, 

are refined in an iterative procedure called NN training, which involves measured track geometry 

and measured vehicle response data. Once the NN is trained, it can be used to predict vehicle 

response from measured track geometry (Haigermoser et al., 2015; Karis, 2018).

Methods based on filters and transfer functions. Haigermoser et al. (2015) and Karis (2018) 

describe a number of other vehicle response analysis methods, including the VRA and Pupil 

method used in the Netherlands, Empirical Transfer Functions (ETF) methods developed in 

Austria, system identification methods developed in Japan, and a number of system identification 

methods developed in Europe under the DYNOTRAIN program. Some of these methods operate 

in the distance domain, others in the frequency or wavelength domain (see Section 7).

6.7 Comparative Assessment of Distance Domain Track Geometry 

Characterization Methods

FRA-sponsored TQI study, 1980s. In the early 1980s, several TQIs were studied as a part of an 

FRA-sponsored program (Hamid and Gross, 1981). The following TQIs were investigated:

• Gauge: mean, SD, 99th percentile, third moment of gauge in in3

, fourth moment of gauge 

in in4

• Cross level: SD of cross level after subtraction of moving mean

• Warp: SD and 99th percentile of 20-ft warp

• Profile: SD of space curve, SD of 3-ft MCO, 99th percentile of 16-ft MCO



62 

for variability in vertical track stiffness along the distance of the track and that more research on 

the use of second derivatives is needed.





68 

FFT, PSD, TF and coherence functions are often used to verify outputs from TGMS (see Section 

5.4). In some cases, frequency analysis of alignment data can reveal missing wavelengths and 

reveal if restoration of space curve data from MCO measurements has not been performed 

correctly (Haigermoser et al., 2013; Grabner, 2013a and 2013b).

7.3 Interpretation and Limitations of PSD

During the transformation from space curve to PSD, data from a large length is averaged and 

phase information is lost. Thus, the PSD does not fully describe the shape of the space curve. For 

example, a PSD of a signal consisting of large discrete random amplitude pulses may be identical 

to a PSD of a signal with small overlapping pulses (Corbin, 1980).

The extent to which a PSD describes track irregularity depends, to a large extent, on their 

statistical nature (see Section 6.4). If, as some studies suggest, track irregularity is well 

approximated by a stationary Gaussian random model, then the peak amplitudes of irregularities 

(and therefore track Class defects) can be sufficiently well predicted from the PSD. If, on the 

other hand, track irregularities are a combination of a continuous stationary random process, a 

periodic deterministic process, and track anomalies, like other studies suggest, then peak 

amplitudes cannot be predicted from the PSD alone.

A continuous stationary random process is adequately described by the PSD continuum (i.e., 

overall shape of the PSD curve). A periodic deterministic process describes uniform track 

irregularities, such as bolted rail joints. This process contributes to both PSD continuum and 

pronounced peaks in the PSD plot; such peaks are located at wavelengths corresponding to rail 

length and its submultiples, i.e., L, L/2, L/3, etc. (see Figure 40 and Figure 41). Track geometry 

anomalies represent isolated events that are not properly characterized by PSDs due to the 

averaging and phase suppression properties of PSDs. The PSD of track with anomalies looks 

similar to the PSD of the same track with anomalies removed (Corbin, 1980).

Figure 40. PSD of alignment, profile, and cross level, showing the effects of continuous 

stationary random process and a periodic deterministic process (Data from the 

DYNOTRAIN project (Haigermoser et al., 2013)) 

70 

China Academy of Railway Sciences (CARS) proposed the following polynomial to describe 

alignment, profile, gauge, and cross level PSDs of different track classes (Chen et al., 2008, as 

cited in Berawi, 2013, and Haigermoser et al., 2015): 

French National Railway Company (SNCF) proposed a polynomial approximation for profile 

PSD (Berawi, 2013; Haigermoser et al., 2015):

It has been suggested (Hamid et al., 1983; Iyengar and Jaiswal, 1995) that the overall shape of 

the PSD curve as characterized by breakpoints does not significantly change with track class. 

Rather, the difference between track classes is characterized by a roughness parameter. In other 

words, PSD plots of different track classes are curves of similar shape, shifted vertically (Figure 

42). On the other hand, data collected in the DYNOTRAIN project shows differences in the 

shape of PSD curves not only between track classes but also between different railway networks 

(Figure 43).

Figure 42. Proposed parametrization for PSD curves corresponding to

different track classes (Hamid et al., 1983)

72 

8. Processing of Track Geometry Data for Vehicle Dynamic 

Simulations

One of the uses of measured track geometry is to generate inputs for MBD simulation of rail 

vehicles. Some of the governmental regulations (49 CFR §213 Appendix D) and industry 

standards for vehicle qualification (EN 14363:2016; AAR MSRP Ch. 11, 2015; AAR M-976, 

2013; AAR S-2043, 2009) prescribe certain requirements for track geometry inputs into MBD 

simulations, such as deterministic track defect shapes (Figure 44), frequency content of track 

irregularities, or standardized track geometry files representing different track classes.

Figure 44. Basic layout of Minimally Compliant Analytical Track (MCAT) (49 CFR §213 

Appendix D) 

Because many details of this processing depend on the MBD software, the procedures described 

below are meant as general guidelines.

8.1 Use of Measured Irregularities

Measured track irregularities are most often used when MBD simulation is used to calibrate a 

vehicle model or to predict the vehicle’s response to a known track segment.

Track geometry data for MBD software must be in a space curve format. If only MCO data is 

available, it must be decolored (Section 5.3), and the user must understand which wavelengths 

cannot be restored and how they may affect vehicle behavior. At a minimum, MBD software 

requires information on track curvature, centerline alignment (i.e., mean alignment), track 

centerline profile (i.e., mean profile/surface), gauge, and superelevation. Alternatively, it may 

require track curvature, left rail alignment, right rail alignment, left rail surface, and right rail 

surface (which is the format usually outputted by TGMS). The conversion between the two 

formats is shown in Equations (8) – (10). 


75 

PSDs can be based on inverse Fourier Transforms, wavelets, or linear dynamic systems; see 

(Haigermoser et al., 2015; Klöckner et al., 2017; Panunzio et al., 2017) for an overview of 

synthesis methods.

If a combined stochastic-deterministic approach is used, shapes of irregularities are deterministic 

(e.g., Equation (49)), but their amplitudes and/or wavelengths are determined stochastically (e.g., 

Equation (50)).
76 

9. Summary and Conclusions

Track geometry measurement is a rapidly developing field. In the last four decades, track 

geometry measurement systems became more affordable, modular, autonomous, and accurate 

over wider ranges of irregularity wavelengths and vehicle speeds.

Track geometry almost always is measured and recorded in the distance domain, with variables 

such as track curvature, alignment, surface, gauge, and cross level plotted against the distance 

along the track (i.e., chainage). This format helps separate track design elements, such as curves 

and spirals, from track irregularities. In the United States track geometry nomenclature is not 

fully standardized, with different agencies sometimes using conflicting or ambiguous terms (e.g., 

cross level vs. superelevation, warp vs. twist).

Alignment and surface can be recorded in the format of space curve or chordal offsets. Chordal 

offsets are widely used because they easily can be verified with hand measurements. Space curve 

is more difficult to measure but, unlike chordal offsets, it does not hide certain defects, does not 

depend on arbitrary values (i.e., chord lengths and versine ratios), and is more directly related to 

the vehicle’s response; thus, it is a more objective method of recording and assessing track 

irregularity data.

One of the difficulties in measuring and processing track geometry data is that different 

conditions call for different processing settings. For example, filter settings that are appropriate 

for FRA Class 1 track may not be appropriate for FRA Class 8 track. European standard EN 

13848 shows an example of a flexible approach in which high-speed tracks require measuring 

track irregularities over wider bandwidth than low-speed tracks.

Another challenge is separation of track irregularities from artifacts caused by special trackwork 

and various environmental conditions. This process is difficult to automate; manual input may be 

required to achieve good sensitivity and specificity.

Track irregularities are characterized in terms of maximum and minimum values of track 

irregularity variables. When these maxima exceed permissible limits for a given track class, they 

are called defects or exceptions. Track irregularities also can be characterized by calculating 

TQIs, which do not only consider the extreme values but also the overall variation of track 

geometry. Many different TQIs have been proposed; most of them are based on standard 

deviations of track irregularity variables and their weighted combinations. Other TQIs based on 

percentiles, first and second derivatives, fractal dimensions, etc., also have been proposed. Some 

agencies use a single index characterizing vertical and lateral irregularities. Studies comparing 

different TQIs in terms of their correlation with vehicle reactions (i.e., accelerations and wheelrail forces) show results that are often contradictory and/or inconclusive.

Which TQI is preferable over others is a difficult question. Several comparative studies 

emphasize the correlation between TQI and FRA Track Class. Indeed, there is some value in 

being able to predict the extreme values of track irregularities from a single TQI value. On the 

other hand, it can be argued that a TQI that is tailored to predict Track Class would be redundant, 

and a better approach is to separate Track Class (as characterized by extreme values of 

irregularities) from track quality (as characterized by SD or other TQIs). This view appears to be 

reflected in the Euronorms, which use maximum irregularity values to define track speed range 

(equivalent to FRA Track Class) and SD values to determine track quality class. For example, a 
77 

track with alignment SD below 0.5 mm is defined as Track Quality Class A if the maximum 

applicable speed is 120 km/h, but the same SD corresponds to Track Quality Class C if the 

maximum applicable speed is 300 km/h (EN 13848–6:2014). 

Characterization of track geometry in wavelength/frequency domain is useful for making general 

conclusions about large sections of track and when validating data from track geometry 

measurement systems. The transition from the distance domain to the wavelength domain is 

usually done with FFT, with results presented as magnitude and phase or PSD plots, which can 

be approximated by piecewise linear functions or polynomials. The relationship between 

different track geometry variables can be explored with CPSD plots, coherence plots, and 

transfer functions. Left and right rail alignment signals tend to have high coherence, especially at 

long wavelengths; the same is true of left and right profile signals. On the other hand, mean 

alignment, mean profile, gauge, and cross level signals have low coherence, i.e., are more or less 

independent. 

Methods for processing measured track geometry into inputs for MBD simulations are well 

established, as are the methods for synthesizing artificial track geometry data.
78 

10. Recommendations for Track Geometry Characterization Study

Based on the results of this literature review, the following approach is recommended for 

characterization of U.S. track geometry data collected within FRA’s ATIP:

1. Space curve data, rather than chordal data, should be used for characterization. This 

creates a challenge for data collected at Track Classes 1-2 (and a high percentage of Class 

3 track), where space curve is unavailable due to slow vehicle speed. This leads to two 

options: 

a. Forfeiting the characterization of alignment and profile on Track Classes 1-3, and 

only characterizing gauge and superelevation for these Track Classes

b. Restoring space curve from chordal data, while being aware of the limitations of 

this process and using caution when comparing the statistics based on restored data 

from Track Class 1-3 with statistics based on the original space curve data from 

higher Track Classes

2. Data should be pre-screened for:

a. Artifacts due to special trackwork and other causes, determined based on first 

spatial derivative exceeding certain threshold (to be determined)

b. Sections where alignment and surface measurements are unreliable due to low 

speed

c. Sections where alignment and surface measurements are inconsistent with gauge 

and superelevation measurements

Data from these sections should not be used in the analysis. Depending on the length of 

the section, data can be either overwritten by interpolating between edge values, or the 

section may be cut out entirely. Additional smoothing or filtering may need to be applied 

to avoid discontinuities. 

3. The following variables should be analyzed: curvature, gauge, superelevation/cross level, 

left alignment, right alignment, centerline alignment, left profile, right profile, centerline 

profile, twist, and warp.

4. Alignment and profile data should be pre-filtered. FIR or cascaded forward-reverse 2nd or 

4th order Butterworth filters appear to be the most likely candidates. Several filters should 

be designed, with exact settings to be determined during analysis:

a. High-pass filter with ~80 ft cutoff wavelength, to be used for lower classes of track 

(similar to D1 domain in Euronorms)

b. High-pass filter with ~200 ft cutoff wavelength, to be used for higher classes of 

track (similar to a combination of D1 and D2 domains)

c. Band-pass filter with ~80 ft high cutoff and ~200 ft low cutoff (similar to D2 

domain) 

d. Low-pass filter with ~200 ft cutoff (similar to D3 domain)
79 

Cutoff wavelength should be refined such that the magnitudes of the filter artefacts in 

spirals are smaller than typical irregularities in the same track classes.

5. Exploratory data analysis should be performed on each space curve variable to determine 

the characteristics of its statistical distribution.

6. Track data should be segmented by track type (i.e., tangent, curve, spiral). When segment 

length exceeds 0.1 mile, it should be segmented into 0.1-mile segments. If locations of 

turnouts and bridges are known, they should be separated into special segments.

7. In the distance domain, the following statistical features should be extracted from the 

space curve variables in each segment:

a. Minimum, maximum, mean, standard deviation, 1st, 5th, 50th, 95th and 99th

percentiles

b. Same statistics of absolute values of space curve variables

c. Minimum, maximum, mean, standard deviation, 1st, 5th, 50th, 95th and 99th

percentiles of first and second spatial derivatives of space curve variables

d. Same statistics of absolute values of derivatives of space curve variables

e. Number of FRA Track Class exceptions in a segment

f. Ratio of combined length of FRA Track Class exceptions to the total segment 

length 

g. Length-based TQI (based on the method by Zhang and El-Sibaie)

h. Other TQIs calculated from these values, if desired

i. Actual FRA Track Class based on exceptions

8. Statistical features from different track segments should be aggregated by track category 

(e.g., FRA Class 4, primarily freight, tangent track). For each track category, distribution 

of previously discussed statistical features (e.g., standard deviations, TQIs, etc.) should be 

presented in terms of probability density distribution and/or cumulative probability 

distribution. 

9. Segmentation of track used for wavelength domain is to be determined. Track segment 

lengths should likely be longer than what is used for distance domain analysis. At the 

same time, calculating one single PSD function based on the entire track category may 

mask meaningful differences (Figure 43). Segment lengths of approximately 0.3-3 miles 

may be ideal for wavelength analysis.

10. In the frequency/wavelength domain, the following should be calculated:

a. PSD of (unfiltered) space curve variables. The exact method is to be determined in 

the process of analysis. FFT or Welch’s method with Hanning windowing are the 

most likely candidates.

b. Coherence between space curve variables (e.g., left and right rail alignment, 

centerline alignment and gauge, gauge and cross level, etc.) 

11. Parametrization of PSD data should be attempted using piecewise linear functions or 

polynomial expressions.
80 

This approach may need to be modified based on the results of the exploratory data analysis and 

other findings throughout the characterization process.
81 

11. References 

1. Ahmadian, M. (1999). Filtering effects of mid-chord offset measurements of track 

geometry data. Proceedings of the 1999 ASME/IEEE Joint Railroad Conference, April 15, 1999. 

2. Aknin, P. & Chollet, H. (1999). A new approach for the modelling of track geometry 

recording vehicles and the deconvolution of versine measurements. IAVSD Conference, Pretoria, 

1999. 

3. Al-Nazer, L. (2014). Track Profile Approximation Using Railcar Body Acceleration Data

(Report No. DOT/FRA/ORD-14/42). Federal Railroad Administration.

4. Andani, M. T. (2016). The application of Doppler LIDAR technology for rail inspection 

and track geometry assessment. PhD Dissertation, Virginia Tech, Blacksburg, Virginia.

5. Andani, M.T., Peterson, A., Munoz, J., & Ahmadian, M. (2018). Railway track 

irregularity and curvature estimation using doppler LIDAR fiber optics. Journal of Rail and 

Rapid Transit, 232(1), 63–72.

6. Anderson, M. (2002). Strategic planning of track maintenance. Department of 

Infrastructure, Borlänge. 

7. Andrews, J., Prescott, D., & De Rozieres, F. (2014). A Stochastic Model for Railway 

Track Asset Management. Reliability Engineering & System Safety, 130(10), 76–84. 

8. Arasteh Khouy, I., Larsson-Kråik, P.O., Nissen, A., & Kumar, U. (2016). Cost-effective 

track geometry maintenance limits. Journal of Rail and Rapid Transit, 230(2), 611–622. 

9. Association of American Railroads, Manual of Standards and Recommended Practices, 

Section C, Part II, Chapter 11: Service-Worthiness Tests and Analyses for New Freight Cars, 

Washington, DC, 2015.

10. Association of American Railroads, Manual of Standards and Recommended Practices, 

Specification M-976: Truck Performance for Rail Cars, Washington, DC, 2013.

11. Association of American Railroads, Manual of Standards and Recommended Practices, 

Standard S-2043: Performance Specification for Trains Used to Carry High-Level Radioactive 

Material, DC, 2009.

12. Bentley Systems, Inc. OpenRail Designer: rail Track and Electrification Design and 

Analysis Software. Accessed January 2, 2019.

13. Berawi, A.R.B. (2013). Improving Railway Track Maintenance Using Power Spectral 

Density. PhD Dissertation, University of Porto.

14. Bracciali, A., Piccioli, F., & De Cicco, T. (2009). Measurement and analysis of mid 

wavelength rail irregularity. Proceedings of Railway Engineering 2009 Conference, London, 

June 2009. 

15. Broquetas, A., Comerón, A., Gelonch, A., Fuertes, J.M., Castro, J. A., Felip, D., López, 

M. A., & Pulido, J.A. (2012). Track Detection in Railway Sidings Based on MEMS Gyroscope 

Sensors. Sensors. 

16. Brown, A.B. & Ashmore, S.J. (2018). Union Pacific Railroad. Personal communication. 

December 4, 2018.

17. Burton, T. (2018). RILA Train Mounted Survey System, Opportunities & Constraints. 
82 

18. Camacho, D., Lee, T.H., Rapp, S., & Martin, U. (2016). Light rail ballasted track 

geometry quality evaluation using track recording car data. Proceedings of the 15th International 

Conference on Railway Engineering Design and Operation (CR 2016). 

19. Carr, G., Diaz, C., & Martin, G. (2002). Determining profile and alignment using an 

optical technique with extrapolations. Proceedings of ASME/IEEE Joint Rail Conference, 

Washington, DC, April 23-25, 2002. 

20. Cerna, M. & Harvey, A.F. (2000). The Fundamentals of FFT-Based Signal Analysis and 

Measurement. National Instruments, Application Note 041, 2000. Accessed: Dec. 17, 2018.

21. Chen, Q., Niu, X., Zhang, Q. & Cheng, Y. (2015). Railway track irregularity measuring 

by GNSS/INS integration. Navigation, 62(1), 83-93. 

22. Chen, Q., Niu, X., Zuo, L., Zhang, T., Xiao, F., Liu, Y., & Liu, J. (2018). A railway track 

geometry measuring trolley system based on aided INS. Sensors, 2018(18). 

23. Chen, X., Wang, L., Tao, X., Cui, G., Yang, F., Chai, X., & Wu, W. (2008). Study on the 

judgement method for track irregularity of the main railway lines in China. Journal of China 

Railway Science, 9(4), 22–27. 

24. Chernov, N. (2009). Circle Fit (Pratt Method). Accessed: March 22, 2019. 

25. Chung, W. & Ham, Y. (2004). Investigation of the track irregularities reconstruction 

methods. Key Engineering Materials, 270-273, 1659-1664. 

26. Ciobanu, C. (2016). The versine formulae. Accessed: December 12, 2018.

27. Clouse, A. (2018). Federal Railroad Administration. Personal communication. April 6, 

2018. 

28. Cohen A. & Hutchens, W. (1970). Methods for the reconstruction of rail geometry from 

mid-chord offset data. Joint Transportation Engineering Conference, Chicago, IL, USA, October 

11-14, 1970. 

29. Corbin, J.C. (1980). Statistical Representations of Track Geometry. Volume I – Main 

Text (Report. No. FRA/ORD-80/22.1). Federal Railroad Administration.

30. Craft, M. (2018). Amtrak. Personal communication. November 29, 2018.

31. Cusumano, J.P. (2005). The Power Spectral Density and the Autocorrrelation. 

Experimental Nonlinear Dynamics: Supplemental Handout. 

32. Davis, T.G. (1999). Total least-squares spiral curve fitting. Journal of Surveying 

Engineering, 125(4).

33. Ebersöhn, W. & Conrad, J.R. (2003). Implementing a railway infrastructure maintenance 

system. Amtrak, Washington, DC.

34. Elkhoury, N., Hitihamillage, L., Moridpour, S., & Robert, D. (2018). Degradation 

Prediction of Rail Tracks: A Review of the Existing Literature. The Open Transportation 

Journal, 2018(12), 88–104. 

35. El-Sibaie, M. & Zhang, Y. (2004). Objective track quality indices. Journal of the 

Transportation Research Board No. 1863, 81-87. 

36. Engstrand, A. (2011). Railway Surveying – A Case Study of the GRP 5000. MS Thesis, 

Royal Institute of Technology, Stockholm.
83 

37. Esveld, C., Jourdain, A., Kaess, G., & Shenton, M.J. (1988). Historic data on track 

geometry in relation to maintenance. Rail Engineering International, Issue 2. 

38. European Committee for Standardization (CEN) (2016). EN 14363:2016, Railway 

applications – Testing for the acceptance of running characteristics of railway vehicles – Testing 

of running behaviour and stationary tests.

39. European Committee for Standardization (CEN) (2008). EN 13848-1:2003+A1: Railway 

applications – Track – Track geometry quality – Part 1: Characterization of track geometry.

40. European Committee for Standardization (CEN) (2016). PrEN 13848-1: Railway 

applications – Track – Track geometry quality – Part 1: Characterization of track geometry.

41. European Committee for Standardization (CEN) (2006). EN 13848-2:2006: Railway 

applications – Track – Track geometry quality – Part 2: Measuring systems – Track recording 

vehicles.

42. European Committee for Standardization (CEN) (2017). EN 13848-5:2017: Railway 

applications – Track – Track geometry quality – Part 5: Geometric quality levels - plain line, 

switches and crossings. 

43. European Committee for Standardization (CEN) (2014). EN 13848-6:2014: Railway 

applications – Track – Track geometry quality – Part 6: Characterisation of track geometry 

quality. 

44. European Rail Research Institute (1989). ERRI Question B 176. Bogies with steered or 

steering wheelsets (Report No. 1). 

45. European Rail Research Institute (1999). ERRI Question C210. Development of methods 

to describe geometric track quality, with the aim of analysing dynamic vehicle behavior (Report 

No. 1: Summary Report). Utrecht, 1999.

46. Fazio, A.E. & Corbin, J.L. (1986). Track quality index for high speed track. Journal of 

Transportation Engineering, 112(1), 46–61.

47. Federal Railroad Administration Office of Railroad Safety (2017). Track and rail & 

infrastructure integrity compliance manual; Volume I, Chapter 3: Track safety standards. 

Automated Track Inspection Program (ATIP) Geometry Car Operation. 

48. Federal Railroad Administration Office of Railroad Safety (2017). Track and rail & 

infrastructure integrity compliance manual; Volume II, Chapter 1: Track safety standards. Track 

safety standards Classes 1 through 5.

49. Federal Railroad Administration Office of Safety (2019). Track Geometry Car System 

Capabilities (Report No. FRA DOTX 217). 

50. Gabara, G. & Sawicki, P. (2018). A New Approach for Inspection of Selected Geometric 

Parameters of a Railway Track Using Image-Based Point Clouds. Sensors, 18(3).

51. Garach, L., de Oña, J., & Pasadas, M. (2014). Mathematical formulation and preliminary 

testing of a spline approximation algorithm for the extraction of road alignments. Automation in 

Construction, 47(2014), 1-9. 

52. Ghasemi, A. & Zahediasl, S. (2012). Normality Tests for Statistical Analysis: A Guide 

for Non-Statisticians. International Jounral of Endocrinology and Metabolism, 10(2).

53. Glaus, R. (2006). The Swiss Trolley – a modular system for track surveying. Siebzigster 

Brand, 70. 
84 

54. Grabner, G. (2013). TrioTRAIN: Track geometry analysis. Frankfurt, Germany. 

55. Grabner, G. (2013). TrioTRAIN: Methods for describing and assessing track geometry. 

Frankfurt, Germany. 

56. Grassie, S. (1996). Measurement of railhead longitudinal profiles: a comparison of 

different techniques. Wear, 191(1996), 245-251. 

57. Gustaffson, F. (1996). Determining the initial states in forward-backward filtering. IEEE 

Transactions on Signal Processing, 44(4), 988–992. 

58. Haigermoser, A., Bezin, Y., Coudert, F., Eickhoff, B., Grabner, G., Kraft, S., Thomas, D., 

& Wolter, U. (2013). DynoTRAIN WP2 Deliverable D2.6 – Final Report on Track Geometry. 

59. Haigermoser, A., Eickhoff, B., Thomas, D., Coudert, F., Grabner, G., Zacher, M., Kraft, 

S., & Bezin, Y. (2014). Describing and assessing track geometry quality. Vehicle System 

Dynamics, 52(Supplement 1).

60. Haigermoser, A., Luber, B., Rauh, J., & Grafe, G. (2015). Road and track irregularities: 

measurement, assessment and simulation. Vehicle System Dynamics, 53(7).

61. Hamid & Gross, A. (1981). Track-Quality Indices and Track-Degradation Models for 

Maintenance-of-Way Planning. 60th Annual Meeting of the Transportation Research Board, 

Washington, DC, January 12–16, 1981. 

62. Hamid, Rasmussen, K., Baluja, M., & Yang, T.-L. (1983). Analytical Descriptions of 

Track Geometry Variations. Volume I – Main Text (Report No. DOT/FRA/ORD-83/03.1). 

Federal Railroad Administration.

63. Hanreich, W., Mittermayr, P., & Presle, G. (2002). Track Geometry Measurement 

Database and Calculation of Equivalent Conicities of the OBB Network. Proceedings of 2002 

AREMA Conference. 

64. Hay, W. (1982). Railroad engineering, 2nd Edition. John Wiley & Sons, New York.

65. Heinzel, G., Rudiger, A., & Schilling, R. (2002). Spectrum and spectral density 

estimation by the Discrete Fourier Transform (DFT), including a comprehensive list of window 

functions and some new flat-top windows. 

66. Higgins, C. & Liu, X. (2018). Modeling of track geometry degradation and decisions on 

safety and maintenance: A literature review and possible future research directions. Journal of 

Rail and Rapid Transit, 232(5), 1385–1397. 

67. Hyslip, J., Trosino, M., & Selig, E. (2002). Fractal Analysis of Track Geometry Data. 

Proceedings of the 2002 AREMA Conference. 

68. International Organization for Standardization (1997). Mechanical vibration and shock – 

Evaluation of human exposure to whole-body vibration – Part 1: General requirements (ISO 

2631-1:1997). 

69. Iyengar, R.N. & Jaiswal, O.R. (1995). Random Field Modeling of Railway Track 

Irregularities. Journal of Transportation Engineering. 121(4).

70. International Union of Railways (UIC) (2005). Leaftet 518: Testing and approval of 

railway vehicles from the point of view of their dynamic behavior – safety – track fatigue – ride 

quality, 3rd Edition. 
85 

71. Jia, C., Xu, W., Wei, L., & Wang, H. (2013). Study of Railway Track Irregularity 

Standard Deviation Time Series Based on Data Mining and Linear Model (Article ID 486738). 

Mathematical Problems in Engineering. 

72. Jia, C., Wei, L., Wang, H., & Yang, J. (2014). Study of Track Irregularity Time Series 

Calibration and Variation Pattern at Unit Section (Article ID 727948). Computational 

Intelligence and Neuroscience. 

73. Jiang, Q., Wu, W., Li, Y., & Jiang, M. (2017). Millimeter Scale Track Irregularity 

Surveying Based on ZUPT-Aided INS with Sub-Decimeter Scale Landmarks. Sensors, 2017(17). 

74. Jovanovic, S. (2004). Railway Track Quality Assessment and Related Decision Making. 

2004 IEEE International Conference on Systems, Man and Cybernetics, The Hague, 

Netherlands, October 10–13, 2004. 

75. Karttunen, K. (2015). Influence of rail, wheel and track geometries on wheel and rail 

degradation. Ph.D. Thesis, Chalmers University of Technology, Göteborg, Sweden, 2015. 

76. Karis, T. (2018). Correlation between Track Irregularities and Vehicle Dynamic 

Response Based on Measurements and Simulations. Licentiate Thesis, Royal Institute of 

Technology, Stockholm, Sweden.

77. Karis, T., Berg, M., Stichel, S., Li, M., Tomas, D., & Dirks, B. (2018). Correlation of 

track irregularities and vehicle responses based on measured data. Vehicle System Dynamics; 

56(6), 967–981. 

78. Klöckner, A., Knoblach, A., & Heckmann, A. (2017). How to shape noise spectra for 

continuous system simulation. Mathematical and Computer Modelling of Dynamical Systems, 

23(3), 284–300. 

79. Kraft, S., Causse, J., & Coudert, F. (2015). An approach for the classification of track 

geometry defects. Proceedings of the 24th Symposium of the International Association for 

Vehicle System Dynamics (IAVSD 2015), Graz, Austria, 17–21 August 2015. 

80. Kraft, S., Causse, J., & Coudert, F. (2018). Vehicle response-based track geometry 

assessment using multi-body simulation. Vehicle System Dynamics, 56(2), 190–220. 

81. Kreye, C., Eissfeller, B., & Ameres, G. (2004). Architectures of GNSS/INS Integrations 

Theoretical Approach and Practical Tests. Institute of Geodesy and Navigation, University FAF 

Munich, Neubiberg, Germany. Accessed Dec. 12, 2018.

82. Krug, G.A. & Madejski, J. (2015). Track maintenance strategies optimization problem. 

Railway Engineering 2015 – 13th International Conference & Exhibition, Edinburgh, UK, 2015. 

83. Krug, G.A. & Madejski, J. (2018). Improving track condition by application of quasi 

cumulative distribution function (QCDF). 5th International Conference on Road and Rail 

Infrastructure, Zadar, Croatia, May 17-19, 2018. 

84. Landgraf, M. (2016). Railway Track Condition: Assessment – Aggregation – Asset 

Management. 11th World Congress on Railway Research, Milan, Italy, May 29 – June 2, 2016. 

85. Lasisi, A. & Attoh-Okine, N. (2018). Principal components analysis and track quality 

index: a machine learning approach. Transportation Research Part C: Emerging Technologies, 

91, 230–248. 

86. Lewis, R. (2011). Track geometry recording and usage: Notes for a lecture to Network 

Rail.
86 

87. Lee, S. (2005). Development of Objective Track Quality Indices (Report No. RR05-01). 

Federal Railroad Administration. 

88. Leica Geosystems AG (2018). ATrack Suite by Leica Geosystems: Automated Track AsBuilt Engineering & Optimization Software. Accessed January 2, 2019.

89. Li, D., Hyslip, J., Sussman, T., & Chrismer, S. (2016). Railway geotechnics. CRC Press, 

Boca Raton, FL.

90. Li, D., Meddah, A., Haas, K., & Kalay, S. (2006). Relating track geometry to vehicle 

performance using neural network approach. Proceedings of the Institution of Mechanical 

Engineers, Part F: Journal of Rail and Rapid Transit 220 p. 273–281. 

91. Li, M., Persson, I., Spannar, J., & Berg, M. (2012). On the quality of second-order 

derivatives of track irregularity for assessing vertical track geometry quality. Vehicle System 

Dynamics, 50(Supplement). 

92. Li, H. & Xiao, T. (2014). Improved generalized energy index method for comprehensive 

evaluation and prediction of track irregularity. Journal of Statistical Computation and 

Simulation, 84(60), 1213–1231. 

93. Lindahl, M. (2001). Track geometry for high-speed railways: A literature survey and 

simulation of dynamic vehicle response. Royal Institute of Technology, Stockholm.

94. Liu, R., Xu, O., Sun, Z., Zou, C., & Sun, Q. (2015). Establishment of track quality index 

standard recommendations for Beijing Metro (Article ID 473830). Discrete Dynamics in Nature 

and Society. 

95. Liu, X., Saat, M., & Barkan, C. (2012). Analysis of causes of major train derailments and 

their effect on accident rates. Journal of the Transportation Research Board, 2289, 154–163. 

96. Liu, Y. & Magel, E. (2007). Performance based track geometry and the track geometry 

interaction map. Proceedings of International Heavy Haul Conference, Specialist Technical 

Session, Kiruna, Sweden, June 11-13, 2007. 

97. Lönnbark, G. (2012). Characterization of Track Irregularities with Respect to Vehicle 

Response. Royal Institute of Technology, Stockholm, Sweden.

98. Luck, T., Eissfeller, B., Kreye C., & Meinke, P. (2001). Measurement of Line 

Characteristics and of Track Irregularities by Means of DGPS and INS. International Symposium 

on Kinematic Systems in Geodesy, Geomatics and Navigation, Banff, Alberta, Canada, June 5-8, 

2001. 

99. Madejski, J. & Grabczyk, J. (2002). Continuous geometry measurement for diagnostics 

of tracks and switches. Proceedings of the International Conference of Switches, Delft, 

Netherlands, 2002. 

100. Mahalakshmi, V. & Joseph, K.O. (2013). GPS Based Railway Track Survey System. 

International Journal of Computer Applications in Engineering Sciences, Volume III, Special 

Issue on National Conference on Information and Communication (NCIC'13).

101. Malone, J. (2007). Performace and testing requirements for portable track geometry 

inspection systems. TCRP Research Results Digest, 83. 

102. Mandal, N.K., Dhanasekar, M., & Sun, Y.Q. (2016). Impact Forces at Dipped Rail Joints. 

Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid 

Transit, 230(1), 271–282. 
87 

103. Marquis, LeBlanc, J., & Tajaddini, A. (2014). Vehicle Track Interaction Safety 

Standards. Proceedings of the 2014 Joint Rail Conference (JRC2014), Colorado Springs, CO, 

April 2-4, 2014. 

104. Di Mascio, P., Di Vito, M., Loprencipe, G., & Ragnoli, A. (2012). Procedure to 

determine the geometry of road alignment using GPS data. Procedia – Social and Behavioral 

Sciences, 53, 1204–1217,. 

105. The MathWorks, Inc. Welch’s Power Spectral Density Estimate. Accessed: March 22, 

2019. 

106. The MathWorks, Inc. Zero-Phase Digital Filtering. Accessed: March 22, 2019.

107. Mauer, L. (1995). Determination of Track Irregularities and Stiffness Parameters with 

Inverse Transfer Functions of Track Recording Vehicles. Vehicle System Dynamics Supplement, 

24(1995), 117–132. 

108. McCrae, J. & Singh, K. (2008). Sketching Piecewise Clothoid Curves. 

EUROGRAPHICS: Workshop on Sketch-Based Interfaces and Modeling. 

109. Meddah, A. (2018). TTCI. Personal communication. November 2018.

110. Moskal, A. & Pastucha, E. (2016). Track and gauge geometry measurements – the 

present and future. Measurement Automation Monitoring, 62(2), 66-71. 

111. Morant, S. (2016). Automating geometry measurement offers real-time benefits. 

International Railway Journal.

112. Morell, J. (2017). Evaluation of the Federal Railroad Administration’s Autonomous 

Track Geometry Measurement System Research and Development Program (Report No. 

DOT/FRA/ORD-17/05). Federal Railroad Administration.

113. Muinde, M.S. (2018). Railway track geometry inspection optimization. M.S. Thesis, 

Luleå University of Technology. 

114. Naganuma, Y. & Yada, T. (2016). Development of truly portable track geometry 

recording trolley and accompanying new measurement principle. Proceedings of the 15th

International Conference on Railway Engineering Design and Operation (CR 2016), 2016. 

115. Nielsen, J., Berggren, E., Lölgen, T., Müller, R., Stallaert, B., & Pesqueux, L. (2013). 

RIVAS Deliverable D2.5: Overview of Methods for Measurement of Track Irregularities

Important for Ground-Borne Vibration. International Union of Railways. 

116. Panunzio, A.M., Puel, G., Cottereau, R., Simon, S., & Quost, X. (2017). Construction of 

a stochastic model of track geometry irregularities and validation through experimental 

measurements of dynamic loading. Vehicle System Dynamics, 55(3), 399–426. 

117. Pedanekar, N.R. (2006). Methods for aligning measured data taken from specific rail 

track sections of a railroad with the correct geographic location of the sections. US Patent No. 

US 7,130,753 B2.

118. Pinter, O. (2012). Using of Trimble® GEDO CE system for absolute track positioning. 

[in Czech], Diploma Thesis, Czech Technical University in Prague, Prague.

119. Pratt, V. (1987). Direct least-squares fitting of algebraic surfaces. SIGGRAPH ’87: 

Proceedings of the 14th annual conference on computer graphics and interactive techniques, 

ACM, New York, NY, USA, 1987. 
88 

120. Puzavac, L., Popović, Z., Lazarević, L. (2012). Influence of Track Stiffness on Track 

Behaviour under Vertical Load. Promet – Traffic & Transportation, 24(5), 405–412. 

121. Rail Accident Investigation Board (RAIB) (2014). Rail Accident Report: Class 

investigation into rail breaks on the East Coast Main Line (Report No. 24/2014).

122. Reedman, M. (2014). Kinky Tracks. Conference on Railway Excellence (CORE-2014), 

Adelaide, Australia, May 5–7, 2014. 

123. Roghani, A., Macciotta, R., & Hendry, M. (2015). Combining track quality and 

performance measures to assess track maintenance requirements. Proceedings of the 2015 Joint 

Rail Conference, San Jose, CA, USA, March 23–26, 2015. 

124. Sadeghi, J. (2010). Development of railway track geometry indexes bases on statistical 

distribution of geometry data. Journal of Transportation Engineering, 136(8).

125. Sadeghi, J., Heydari, H., & Doloei, E. A. (2017). Improvement of Railway Maintenance 

Approach by Development of a New Railway Condition Index. J. Transportation Engineering, 

Part A: Systems, 143(8).

126. Scanlan, K.M., Hendry, M.T., & Martin, C.D. (2016). Evaluating the equivalency 

between track quality indices and minimum track geometry threshold exceedances along a 

Canadian freight railway. Proceedings of the 2016 Joint Rail Conference, Columbia, SC, USA, 

April 12–15, 2016. 

127. Schubert, E. (1970). Die Planung von Gleisdurcharbeitungen auf Grund von

Oberbaumessungen. Der Eisenbahningenieur, 21,140–143. 

128. Sharma, S. (2016). Data-Driven Optimization of Railway Track Inspection and 

Maintenance Using Markov Decision Process. M.S. Thesis, State University of New York at 

Buffalo. 

129. Sherrock, E. (2018). ENSCO. Personal communication. Dec. 17, 2018.

130. Smith, S.W. (1997). The Scientist and Engineer’s Guide to Digital Signal Processing. 

California Technical Publishing, San Diego.

131. Soleimanmeigouni, I., Ahmadi, A., & Kumar, U. (2018). Track geometry degradation 

and maintenance modelling: A review. Journal of Rail and Rapid Transit, 232(1), 73–102. 

132. Soleimanmeigouni, I., Ahmadi, A., Arasteh Khouy, I., & Letot, C. (2018). Evaluation of 

the effect of tamping on the track geometry condition: A case study. Journal of Rail and Rapid 

Transit, 232(2), 408–420. 

133. Stow, J. & Andersson, E. (2006). Field testing and instrumentation of railway vehicles. 

Handbook of railway vehicle dynamics, ed. S. Iwnicki, CRC Press, Boca Raton, FL.

134. Stratakos, J., Gikas, V., & Fragos, K. (2009). A multi-scale curve matching technique for 

the assessment of road alignment using GPS/INS data. 6th International Symposium on Mobile 

Mapping Technology, Presidente Prudente, São Paulo, Brazil, July 21–24, 2009. 

135. Stuart (2017). Autonomous track geometry measurement system: technical development 

and short line demonstration. Rail Infrastructure and Vehicle Inspection Technology Conference, 

University of Illinois Urbana-Champaign, June 20-21, 2017. 

136. Sun, Y.Q., Spiryagin, M., Cole, C., & Simson, S. (2013). Effect of Wheel-Rail Contacts 

and Track Gauge Variation on Hunting Behaviours of Australian Three-Piece Bogie Wagon. 
89 

Proceedings of the 23rd International Symposium on Dynamics of Vehicles on Roads and on 

Tracks, Qingdao, China, 19–23 August, 2013. 

137. Sundaram, N. & Wilson, R. (2016). Portable track geometry measurement system: a 

unique derailment investigation tool. Presentation, 2016 APTA Rail Conference, 2016. 

138. Szwilski, T.B., Begley, R., Dailey, P., Sheng, Z., & Rahall, N.J. (2003). Determining rail 

track movement trajectories and alignment using HADGPS. Proceedings of the AREMA Annual 

Conference, Chicago, Ill, USA, October 2003. 

139. Tong, X. & Ding, K. (2010). Estimating Geometric Parameters of Highways and 

Railways Using Least-Squares Adjustment. Survey Review, 42(318), 359–374. 

140. Talukdar, K., Arulomozhi, U., Prrabhakar Reddy, K., & Satyanarayan (2006). 

Improvement of TGI Value by Computer Analysis. Presentation, Indian Railway Institute of 

Civil Engineering. 

141. Transport Canada (2012). TC E-54: Rules Respecting Track Safety. Effective May 25, 

2012. 

142. Trehag, J., Handel, P., & Ogren, M. (2008). Onboard estimation and classification of 

railroad curvature. IEEE Transactions on Instrumentation and Measurement, 59(3).

143. Trimble Inc. (2017). Trimble adds inertial-based trolley solution to its track survey and 

scanning rail portfolio. Press Release. 

144. United States Code of Federal Regulations. 49 CFR §213: Track Safety Standards, 

Subpart C. Track Geometry.

145. United States Code of Federal Regulations. 49 CFR §213: Track Safety Standards, 

Subpart G. Train Operations at Track Class 6 and Higher.

146. United States Code of Federal Regulations. 49 CFR §213 Appendix D: Minimally 

Compliant Analytical Track (MCAT) Simulations Used for Qualifying Vehicles to Operate at 

High Speeds and at High Cant Deficiencies. 

147. Vogelaar, J. (2017). Absolute track geometry, what is it and how does it help me? Rail 

Infrastructure and Vehicle inspection Technology Conference, University of Illinois UrbanaChampaign, June 20-21, 2017. 

148. Wang, P., Wang, Y., Tang, H., Gao, M., Chen, R., & Xu, J. (2018). Error theory of 

chord-based measurement system regarding track geometry and improvement by high frequency 

sampling. Measurement, 115, 204–216. 

149. Wang, Y., Wang, P., Wang, X., & Liu, X. (2018). Position synchronization for track 

geometry inspection data via big-data fusion and incremental learning. Transportation Research 

Part C: Emerging Technologies, 93, 544–565. 

150. Weisstein, E.W. Curvature. MathWorld – A Wolfram Web Resource. Accessed: 

December 12, 2018.

151. Weston, P., Roberts, C., Yeo, G., & Stewart, E. (2015). Perspectives on railway track 

geometry condition monitoring from in-service railway vehicles. Vehicle System Dynamics, 

53(7), 1063–1091. 

152. Wickramarachi, P. (2003). Effects of windowing on the spectral content of a signal. 

Sound and Vibration, 37(1), 10–13.
90 

153. Wolf, G. (2015). Principles of Track Geometry. Presentation, WRI-2015, Principles 

Course, May 19, 2015. 

154. Wrobel, S. (2013). Multi-function LIDAR sensors for non-contact speed and track 

geometry measurement in rail vehicles. M.S. Thesis, Virginia Polytechnic Institute and State 

University, Blacksburg, VA.

155. Xu, P., Sun, Q., Liu, R., & Souleyrette, R.R. (2016). Optimal match method for milepoint 

postprocessing of track condition data from subway track geometry cars. Journal of 

Transportation Engineering, 142(8).

156. Xu, P., Sun, Q., Liu, R., Souleyrette, R.R., & Wang, F. (2014). Optimizing the Alignment 

of Inspection Data from Track Geometry Cars. Computer-Aided Civil and Infrastructure 

Engineering, 30(1).

157. Yada, T., Soda, Y., & Naganuma, Y. (2017). Improvement of the realignment 

performance for short wavelength track irregularity on a tamping machine. WIT Transactions on 

Engineering Sciences, 118. 

158. Yazawa, E. & Takeshita, K. (2002). Development of measurement device of track 

irregularity using inertial mid-chord offset method. Quarterly Report of RTRI, 43(3).

159. Zarembski, A.M., Grissom, G.T., Euston, T.L., & Cronin, J.J. (2015). Relationship 

between missing ballast and development of track geometry defects. Transportation 

Infrastructure Geotechnology, 2, 167–176. 

160. Zhang, Y., El-Sibaie, M., & Lee, S. (2004). FRA track quality indices and distribution 

characteristics. Proceedings of AREMA Annual Conference, Nashville, TN, September 19-22, 

2004. 
91 

Appendix A. 

FRA Track Safety Limits (49 CFR §213)

Table 4. Alignment exception limits defined in United States Code of Federal Regulations 

(49 CFR §213)

Parameter

(all exceptions in 

inches)

Type of 

track

Additional 

qualifiers

Track Class

Exc. 1 2 3 4 5 6 7 8 9 

Operating Speed Limit, Freight (mph) 10 10 25 40 60 80

Operating Speed Limit, Passenger (mph) 15 30 60 80 90 110 125 160 220

Min. gauge 56 56 56 56 56 56 56 56 561⁄4

Max. gauge 581⁄4 58 573⁄4 573⁄4 571⁄2 571⁄2 571⁄4 571⁄4 571⁄4 571⁄4

Change of gauge 

within 31’ 3⁄4 1⁄2 1⁄2 1⁄2 

31’ MCO Tangent 1⁄2 1⁄2 1⁄2 1⁄2 

31’ MCO Curved 11⁄4 1 1⁄2 1⁄2 1⁄2 1⁄2 1⁄2 

62’ MCO Tangent 5 3 13⁄4 11⁄2 3⁄4 3⁄4 3⁄4 3⁄4 1⁄2 

62’ MCO Curved 5 3 13⁄4 11⁄2 5⁄8 5⁄8 1⁄2 1⁄2 1⁄2 

124’ MCO Tangent 11⁄2 11⁄4 1 3⁄4 

124’ MCO Curved 11⁄2 11⁄4 3⁄4 3⁄4 

31’ MCO Curved Outside rail

in operations 

at a qualified 

cant 

deficiency 

>5”

3⁄4 3⁄4 1⁄2 1⁄2 1⁄2 1⁄2 1⁄2 

62’ MCO Curved 11⁄4 11⁄4 11⁄4 7⁄8 5⁄8 5⁄8 1⁄2 1⁄2 1⁄2 

124’ MCO Curved 11⁄4 1 3⁄4 3⁄4 

31’ MCO

For ≥3 nonoverlapping 

deviations 

within a 

distance 5x 

chord 

length, each 

of which 

exceeds 

these limits

3⁄8 3⁄8 3⁄8 3⁄8

62’ MCO 1⁄2 3⁄8 3⁄8 3⁄8

124’ MCO 1 7⁄8 1⁄2 1⁄2 
92 

Table 5. Profile exception limits defined in United States Code of Federal Regulations (49 

CFR §213)

Parameter

(all exceptions in 

inches)

Type of 

track

Additional 

qualifiers

Track Class

1 2 3 4 5 6 7 8 9 

Operating Speed Limit, Freight (mph) 10 25 40 60 80

Operating Speed Limit, Passenger (mph) 15 30 60 80 90 110 125 160 220 

31’ runoff at the 

end of a raise 31⁄2 3 2 11⁄2 1 

31’ MCO 1 1 3⁄4 1⁄2 

62’MCO 3 23⁄4 21⁄4 2 11⁄4 1 1 1 3⁄4 

124’ MCO 13⁄4 11⁄2 11⁄4 1 

Deviation from zero 

cross level Tangent 3 2 13⁄4 11⁄4 1 1 1 1 1 

Reverse elevation Curved 3 2 13⁄4 11⁄4 1 1⁄2 1⁄2 1⁄2 1⁄2 

Superelev.+cross 

level Curved 8 8 7 7 7 7 7 7 7 

62’ warp

Except in some 

spirals 

designed prior 

to 1998

3 21⁄4 2 13⁄4 11⁄2 11⁄2 11⁄2 11⁄4 1 

62’ warp

6 consecutive 

pairs of joints 

≥10’ apart

11⁄4 11⁄4 11⁄4 11⁄4 1 1 1 1 

62’ warp Curved Where 

elevation >6” 11⁄2 11⁄2 11⁄2 11⁄2 11⁄2

10’ warp Curved 11⁄4 11⁄8 1 3⁄4 

31’ warp Spiral

Certain spirals 

designed prior 

to 1998

2 13⁄4 11⁄4 1 3⁄4 

31’ MCO

For operations 

at a qualified 

cant deficiency 

>5”

1 1 1 

62’ MCO 21⁄4 21⁄4 13⁄4 11⁄4 1 

124’ MCO 11⁄2 11⁄4 11⁄4 1 

10’ warp Curved 2 2 13⁄4 13⁄4 11⁄2 11⁄4 1 1 3⁄4 

10’ warp Curved 

For operations 

at a qualified 

cant deficiency 

>7”

11⁄4 1 3⁄4 3⁄4 

31’ MCO

For ≥3 nonoverlapping 

deviations 

within a 

distance 5x 

chord length, 

each of which 

exceeds these 

limits

3⁄4 3⁄4 1⁄2 3⁄8

62’ MCO 3⁄4 3⁄4 3⁄4 1⁄2 

124’ MCO 11⁄4 1 7⁄8 5⁄8
93 

Appendix B.

Proposed Descriptions of Deterministic Track Anomalies (Hamid et al., 1983)

Table 6. Proposed formulae and parameter values for track anomalies (adapted from Hamid et al., 1983)

Name Shape Formulae

Range of Values

Gage Alignment Cross level Profile

A 

(inch) k (ft-1) A (inch) k (ft-1) A (inch) k (ft-1) A (inch) k (ft-1) 

Cusp 0.8–

1.4

0.016–

0.061

0.5–

3.0

0.011–

0.103

0.9–

3.0

0.031–

0.095

0.9–

3.0

0.016–

0.095

Bump 0.8–

1.4

0.031–

0.040

0.5–

2.8

0.009–

0.083

1.0–

3.0

0.017–

0.031

0.5–

4.0

0.013–

0.065

Jog N/A N/A 0.5–

3.3

0.006–

0.025

1.6–

2.8

0.020–

0.050

0.5–

5.0

0.008–

0.045

Plateau 0.8–

1.3

0.029–

0.08

1.2–

1.6

0.025–

0.027

0.6–

1.0 0.026–0.04 0.9–

3.0

0.009–

0.033
94 

Name Shape Formulae

Range of Values

Gage Alignment Cross level Profile

A 

(inch) k (ft-1) A (inch) k (ft-1) A (inch) k (ft-1) A (inch) k (ft-1) 

Trough N/A N/A 1.4–

2.2

0.013–

0.029 N/A N/A 0.7–

2.0

0.020–

0.025

Sinusoid N/A N/A 0.8–

1.2

0.033–

0.020 N/A N/A 1.0–1.5

0.020–

0.025

Damped 

Sinusoid

0.5–

1.0 N/A 1.0–

2.2

0.013–

0.015

0.9–

1.2

0.051–

0.061 N/A N/A

sin(x)/x N/A N/A N/A N/A N/A N/A 1.0–

1.2

0.031–

0.033
95 

Abbreviations and Acronyms

ACRONYM DEFINITION

AAR Association of American Railroads

ATGMS Autonomous Track Geometry Measurement System

ATIP Automated Track Inspection Program

CFR Code of Federal Regulations

CoSD Combined Standard Deviation

CPSD Cross Power Spectral Density

CTR Composite Track Record

DFT Digital Fourier Transform

EN Euronorm

ERRI European Rail Research Institute

ETF Empirical Transfer Function

FFT Fast Fourier Transform

FIR Finite Impulse Response

FRA Federal Railroad Administration

GEI Generalized Energy Index

GNSS Global Navigation Satellite System

GPS Global Positioning System

GQI Gauge Quality Index

IIR Infinite impulse Response

IMS Inertial Measurement System

ISO International Organization for Standardization

LIDAR Light Detection and Ranging

L/V Lateral-to-Vertical

LVDT Linear Variable Differential Transformer

MBD Multi-Body Dynamics

MCO Mid-Chord Offset

MSRP Manual of Standards and recommended Practices

NN Neural Network

OTGI Overall Track Geometry Index
96 

ACRONYM DEFINITION

PCA Principal Component Analysis

PMA Point-Mass Acceleration

P-P Probability-Probability

PSD Power Spectral Density

QCDF Quasi Cumulative Distribution Function

Q-Q Quantile-Quantile

RFID Radio-Frequency Identification

SD Standard Deviation

SQI Surface Quality Index

TF Transfer Function

TGI Track Geometry Index

TGIM Track Geometry Interaction Map

TGIMP Track Geometry Interaction Map Parameter

TGMS Track Geometry Measurement System

TQC Track Quality Class

TQI Track Quality Index

TSI Technical Specifications for Interoperability

TTCI Transportation Technology Center, Inc.

UIC International Union of Railways (Union Internationale des Chemins de 

fer) 

UPRR Union Pacific Railroad

VRA Vehicle Response Analysis

VRMS Vehicle Response Measurement System

VTI Vehicle-Track Interaction